SIMULATION OF THE RUPTURE OF CERAMICS FROM SURFACE FLAWS

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SIMULATION OF THE RUPTURE OF CERAMICS FROM SURFACE FLAWS

J. Jouin, J. Chermant, F. Osterstock

To cite this version:

J. Jouin, J. Chermant, F. Osterstock. SIMULATION OF THE RUPTURE OF CERAMICS FROM SURFACE FLAWS. Journal de Physique Colloques, 1986, 47 (C1), pp.C1-601-C1-605.

�10.1051/jphyscol:1986191�. �jpa-00225622�

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Colloque C1, suppl6ment au n"2, Tome 47, f6vrier 1986 page c1-601

SIMULATION OF THE RUPTURE OF CERAMICS FROM SURFACE FLAWS

J.M. JOUIN, J.L. CHERMANT and F. OSTERSTOCK

Equipe Materiaux-Microstructure du L.A. 251, ISMRa, ~niversit6.

F-14032 Caen Cedex, France

Rdsumi - Dans ce travail nous simulons la propagation de petits dCfauts de surface dans le cas de la fatigue dynamique de c&ramiques polycristallines.

Les parametres essentiels sont un facteur critique dlintensit& de contrainte effectif local et la corrosion sous contrainte. On montre que la contrainte h la rupture reste constante sur un large domaine de tailles du difaut initial. Les rCsultats obtenus sont discut&s en fonction des donnCes de la 1ittCrature. I1 semble que les d6fauts dqindentation sont des dCfauts mod&- les pour de tels travaux, H lqopposC des difauts naturels oh le voisinage microstructural influe beaucoup sur le choix des parametres h introduire dans le programme de simulation.

Abstract - In this work the propagation of small surface flaws in polycrys- talline ceramics has been computer-simulated in the case of dynamic fatigue.

Local effective critical stress intensity factors and subcritical crack growth have been taken into account. It is shown that the rupture stress remains constant with varying initial flaw size in a large size domain. The results are compared and discussed with respect to literature data. Con- trolled flaws from indentation may be used as model flaws for such investi- gation in contrast to natural flaws where the adjacent microstructure af- fects very much the input-parameters for simulation.

I - INTRODUCTION

In recent years it has become apparent that small flaws propagate under critical stress intensity factors lower than those measured with large cracks in specimens (SENB, DCB, DT ... ^{) .} In this respect, simulation with local effective critical stress intensity factors has yielded interesting results in time to rupture studies / I / . H.P. Kirchner et a1 / 2 , 3 / noticed also that the decrease of rupture stress with increasing flaw size is lower than expected from a constant KIC value. They attributed that discrepancy to a monocrystalline-polycrystalline transition. R.W.

Rice et a1 / 4 , 5 / proposed a smoother transition at a given flaw size, a, over grain size, D, ratio : a = a/D. T. Okada and G. Sines / 1 , 6 , 7 / observed a quite good correlation between fractographically measured flaw size and those calculated using simulation in the case of time- to-failure tests.

The aim of this paper is to present results obtained by simulating the growth of small surface flaws subjected to sub-critical growth (stress corrosion). Only the propagation of semi-elliptical flaws will be considered. The interaction of growing adjacent flaws, through an interaction factor ( 8 ) , as used by T. Okada and G.

Sines, will only be introduced for comparison purposes in some cases. The results will be discussed with respect to experimental data from the literature.

Simulation background

The local effective critical stress intensity factor, ~:~~(a), as proposed by R.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986191

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Rice and u s e d by T. Okada and G. S i n e s w r i t e s :

~ : ~ ~ ( a ) = KC[l + (K;/KC - ^{1 ) ( 1 +} c o s n a / a D ) / 2 1

P P

where KC c o r r e s p o n d s t o t h e t o u g h n e s s o f t h e p o l y c r y s t a l l i n e m a t e r i a l s and KC t o t h a t o f P t h e s i n g l e c r y s t a l . The v a r i a t i o n o f K : ~ ~ a s a f u n c t i o n o f f l a w s i z e , g , i s shown i n f i g u r e 1 f o r two v a l u e s of a.

The s u b c r i t i c a l c r a c k g r o w t h , d u e t o s t r e s s c o r r o s i o n , i s t h u s :

where K is t h e a p p l i e d s t r e s s i n t e n s i t y f a c t o r (K = UY&). I n t h e c a s e o f i n t e r a c - t i n g f l a w s we would h a v e K = z ~ Y & . Developing and r e a r r a n g i n g t h e p r e v i o u s equa- t i o n y i e l d s :

T h i s i n t e g r a l e q u a t i o n i s s o l v e d a s f o l l o w s : a t t i m e t , we have :

where o is t h e s t r e s s r a t e .

The i n s t a n t a n e o u s growth r a t e i s t h u s :

d a / d t =' V{t) = A C K ~ ( ~ ) / K : ~ ~ ( a l l N S o , f o r t h e n e x t s t e p , we t a k e : a ( t + At) = a c t ) + V ( t ) A t .

The i n c r e m e n t a l t i m e s t e p , A t , i s c h o s e n s u c h t h a t : A o = = 0.01 MPa.

I n p u t d a t a a r e t h o s e ; o r a commercial alumina ( 9 6 % d e n s i t y and 32 pm mean g r a i n s i z e ) : KC = 5.3 MPa -:n, KC = 1 . 7 MPa 6, N = 3 1 and Y = 1.27. F r a c t u r e o c c u r s when K ( t ) = P ~ C ( a ) , i . e . V ='A. We h a v e c h o s e n A = m / s e c , which seems p h y s i c a l l y

I .

meanlngful'f$hese d a t a a r e t h e same a s t h o s e of T. O k d a and G. S i n e s 111.

oal Y

F i g . 1 - S c h e m a t i c f o r t h e v a r i a t i o n o f t h e l o c a l e f f e c t i v e c r i t i c a l s t r e s s i n t e n - s i t y f a c t o r , KZff, between K' and Kc. Two v a l u e s f o r a h a v e b e e n u s e d .

P RESULTS

F i g u r e 2 d e s c r i b e s t h e g e n e r a l f e a t u r e s o f s u c h a c r a c k p r o p a g a t i o n s i m u l a t i o n . They a r e r e p r e s e n t e d w i t h i n logK - logV c o o r d i n a t e s b u t i n o r d e r t o a v o i d c o n f u s i o n w i t h K - V diagrams from l a r g e s p e c i m e n s and m a c r o c r a c k s (DT, DCB, ... ^{) , t h e y}

s h o u l d 'be d e n o t e d a s c r a c k growth p a t h s . The i n s t a n t a n e o u s v a l u e s of N a r e a l s o g i v e n . Some remarks c a n be made from t h e o b s e r v a t i o n o f f i g u r e 2. Depending o n t h e s i z e of t h e i n i t i a l f l a w , it e i t h e r s t a b i l i z e s b e f o r e r u p t u r e o r becomes u n s t a b l e b e f o r e KC h a s been r e a c h e d . T h i s h a s a l s o been o b s e r v e d by T. Okada and G. S i n e s . The ~ - 1 e 8 e l o f c r a c k s t a b i l i z a t i o n depends among o t h e r s on t h e v a l u e of N , t h a t of t h e s t r e s s r a t e , o r t h a t o f a,....

0 0.2 0.4 0.6 Log K- 0 10 20 30 40 50 60 70 80 aopm 0 F i g . 2 - S i m u l a t e d c r a c k growth F i g . 3 - V a r i a t i o n o f t h e r u p t u r e s t r e s s p a t h s f o r v a r i o u s i n i t i a l f l a w s i - w i t h t h e i n i t i a l f l a w s i z e , a . S e v e r a l z e s . O t h e r e x p e r i m e n t a l i n p u t p a r a - c r i t e r i a ( l o c a l , p o l y c r y s t a l l i g e , i n t e r - m e t e r s a r e a l s o g i v e n . a c t i o n ) a r e used h e r e .

The e f f e c t of b o t h t h e i n t e r a c t i o n p a r a m e t e r , 2, and t h e e f f e c t i v e l o c a l c r i t i c a l s t r e s s i n t e n s i t y f a c t o r ,

~ z ~ ~ ,

As soon a s s u b c r i t i c a l c r a c k growth i s i n t r o d u c e d , t h e l o c a l f r a c t u r e c r i t e r i o n , K : ~ ~ i s changed. T h i s i s shown i n f i g u r e 4 where t h e v a r i a t i o n o f t h e l o c a l K i s drawn i n t h e c a s e s : N = 31, u = 6 , P = 160 pm w i t h o r w i t h o u t t a k i n g i n t o acEount t h e f a c t o r s Z ( a ) and K f f ( a ) . For t h a t l e v e l of i n i t i a l f l a w s i z e , t h e c r i t i c a l d e f e c t s i z e , a , l e a % t o K e f f and ⁰ . T h i s g i v e s t o t h e c u r v e shown t h e same f e a t u r e , i . e . t h z e x i s t e n c e of a platea: ( F i g . 5).

F i g . 4 - I n f l u e n c e o f s t r e s s c o r r o s i o n ( s u b c r i t i c a l c r a c k g r o w t h ) o f t h e v a r i a t i o n of t h e l o c a l e f f e c t i v e c r i t i c a l s t r e s s i n t e n s i t y f a c t o r ( t o compare w i t h f i g u r e 1 ) .

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F i g . 5 - V a r i a t i o n o f s t r e s s and f l a w s i z e F i g . 6 - V a r i a t i o n of K : ~ w i t h i n i - a t r u p t u r e a s a f u n c t i o n of i n i t i a l s i z e , t i a l f l a w s i z e and two v a f u e s f o r a.

a , f o r t h e same i n p u t p a r a m e t e r s a s i n O t h e r p a r a m e t e r s a r e a s i n f i g u r e 4.

f P g u r e 4 . Reducing (Y (example of a l a r g e c r y s t a l

encompassed by s m a l l g r a i n s w i t h r e s - p e c t t o t h e mean) r e d u c e s t h e K - l e v e l and t h e w i d t h o f t h e p l a t e a u .

DISCUSSION AND CONCLUSION

The main f i n d i n g o f t h a t s i m u l a t i o n work i s t h a t t h e r e e x i s t a p l a t e a u o f c o n s t a n t 0 - v a l u e s o v e r a l a r g e r a n g e of i n i t i a l f l a w s i z e s . When s e v e r a l n e a r b y f l a w s a r e

&owing s i m u l t a n e o u s l y , t h e i r c o a l e s c e n c e i s t h e f i n a l s t a g e of f r a c t u r e and t h e t e n d a n c y toward a p l a t e a u o f c o n s t a n t a - v a l u e s becomes l e s s pronounced ( F i g . 3 ) . The i n f l u e n c e of t h e i n t e r a c t i o n p a r a m e t g r w i l l b e a n a l y s e d i n f u r t h e r work.

The p r e s e n c e of a p l a t e a u seems t o b e i n c o n t r a d i c t i o n w i t h u s u a l e x p e r i m e n t a l r e s u l t s i n t h e f i e l d of thermomechanical c e r a m i c s . The s c a t t e r of t h e r u p t u r e s t r e s s e s i s w e l l known. I n t h a t f i e l d t h e W e i b u l l s t a t i s t i c ( w e a k e s t l i n k t h e o r y ) h a s been t h e most u s e d b e c a u s e i t was founded on t h e a s s u m p t i o n t h a t t h e r e e x i s t s a d i r e c t and s i m p l e c o r r e l a t i o n between t h e f l a w s i z e , a , and t h e r u p t u r e s t r e s s , a i .e. q ^O: KIC/&, where KIC i s c o n s i d e r e d t o b e a c o n s t a n t . r '

- N a t u r a l f l a w s

I n t r o d u c i n g t h e l o c a l e f f e c t i v e v a l u e o f t h e s t r e s s i n t e n s i t y f a c t o r , ~ z ~ ~ ( a ) , c o u l d e x p l a i n t h e r e s u l t s o f H.P. K i r c h n e r e t a 1 / 3 / i . e . t h e r u p t u r e s t r e s s d e c r e a s e s much l e s s a s e x p e c t e d when t h e f l a w s i z e i n c r e a s e s . E f f e c t i v e l y t h e s m a l l f l a w s become i n s t a b l e a t much lower K - v a l u e s t h a n l a r g e o n e s . On t h e o t h e r hand i f one o b s e r v e d t h e r e s u l t s o f S.G. S e s h a d r i and M. S r i n i v a s a n 1 9 1 , t h e s c a t t e r of t h e C a - r e s u l t s c a n n o t b e r e d u c e d by u s i n g two d i f f e r e n t f r a c t u r e c r i t e r i a .

T h i s may be u n d e r s t o o d i n terms of t h e m i c r o s t r u c t u r a l d i v e r s i t y of e a c h f l a w . Such n a t u r a l f l a w s , i . e . p o r e s , l a r g e c r y s t a l s , a g g r e g a t e s , may h a v e v a r i o u s m i c r o s t r u c - t u r a l s u r r o u n d i n g s . A l a r g e p o r e h a s f r e e s u r f a c e s and t h u s s p e c i f i c t h e r m a l expan- s i o n r e s i d u a l s t r e s s e s . A l a r g e c r y s t a l i s m o s t l y encompassed by s m a l l c r y s t a l s a n d , u n l i k e a c l u s t e r of s i m i l a r c r y s t a l s , h a s a l s o o t h e r r e s i d u a l t h e r m a l s t r e s - s e s . I n t h e v i c i n i t y o f e a c h i n h o m o g e n e i t y , t h e c h e m i c a l c o m p o s i t i o n o f t h e g r a i n b o u n d a r i e s i s l i k e l y t o b e d i f f e r e n t from t h e b u l k of t h e m a t e r i a l . As a consequen- c e e a c h f l a w o r t y p e o f f l a w s w i l l h a v e i t s own s e t of (A, N , ^ci ) v a l u e s and t h e p l a t e a u s of c o n s t a n t a - v a l u e s w i l l b e s h i f f t e d . F i g u r e 6 g i v e s t h e r e s u l t s of v a r y i n g t h e v a l u e o f a ;'the s h i f t o f t h e c o n s t a n t a - p l a t e a u i s w e l l d e f i n e d . T h a t c a s e would w e l l d e s c r i b e a l a r g e c r y s t a l s u r r o u n d e d gy s m a l l c r y s t a l s . Taking i n t o a c c o u n t t h e l o c a l t h e r m a l r e s i d u a l s t r e s s e s i n s u c h a c a s e would need a d d i t i o n a l i n v e s t i g a t i o n .

- I n d e n t a t i o n f l a w s

The s i m u l a t i o n r e s u l t s c a n a l s o b e a p p l i e d t o c o n t r o l l e d f l a w t e s t i n g . N a t u r a l f l a w s a r e v e r y l o c a l i z e d and i f t h e y provoke f r a c t u r e , t h e i r m i c r o s t r u c t u r a l v i c i - n i t y w i l l d e f i n e t h e f r a c t u r e p a r a m e t e r s a s a f u n c t i o n o f t h e inhomogeneity o f t h e m a t e r i a l . H a r d n e s s i n d e n t a t i o n s a f f e c t t h e b u l k i n a random way and have t h u s a v e r y s m a l l p r o b a b i l i t y t o b e l o c a l i s e d n e a r m i c r o s t r u c t u r a l i n h o m o g e n e i t i e s . A s a consequence t h e y may be r e g a r d e d a s model-flaws, a s used i n s i m u l a t i o n .

With r e g a r d t o t h e p r e v i o u s comments, t h e c o n t r o l l e d f l a w r e s u l t s of S.G. S e s h a d r i and M. S r i n i v a s a n / 9 / and t h o s e of B.R. Lawn e t a l . / l o / may be c o n s i d e r e d a s p a r t i c u l a r examples f i t t i n g v e r y w e l l t o o u r r e s u l t s .

S.G. S e s h a d r i and M. S r i n i v a s a n / 9 / u s e d t h e c r a c k p r o p a g a t i o n c r i t e r i a o f G.R.

I r w i n / 1 0 , 1 1 / and F.R. B a r a t t a 1121 t o c o r r e l a t e t h e f r a c t u r e s t r e s s w i t h t h e e q u i v a l e n t f l a w s i z e s ( r e s p e c t i v e l y f l a t s e m i - e l l i p t i c a l and t h r e e d i m e n s i o n a l v o i d w i t h a c i r c u m f e r e n t i a l c r a c k models) o n s i n t e r e d a - S i c . The h i g h e s t f r a c t u r e s t r e s - s e s had been measured w i t h i n t r i n s i c o r n a t u r a l f l a w s . T h e i r s c a t t e r h a s a l r e a d y b e e n d i s c u s s e d . To e x t e n d t h e r a n g e o f f l a w s i z e , t h e y a l s o u s e d t h e c o n t r o l l e d f l a w t e c h n i q u e ( V i c k e r s i n d e n t a t i o n ) . T h e i r s i z e v a r i e d from 120 um t o 250 pm.

F i r s t t h e K - v a l u e s from c o n t r o l l e d f l a w s were lower t h a n t h o s e measured w i t h SENB specimens (KIC C 4.5 t o 5 M P ~ & ) . Secondly t h e y v a r i e d from 2 M P ~ & t o 3 M P ~ & a s t h e i n d e n t a t i o n f l a w s i z e s i n c r e a s e d from 120 pm t o 250 pm. On t h e 0 v s . f l a w s i z e p l o t t h i s r e s u l t s i n a p l a t e a u . T h i s i s i n q u a l i t a t i v e agreement w i t 6 t h e l e f t - h a n d p a r t o f t h e u v s a v a r i a t i o n i n f i g u r e 5.

The r i g h t - h a n d p a r t o f t h a t c u r v e c p l a t e a u + d e c r e a s e o f 0 ) may be c o n f i r m e d by t h e e x p e r i m e n t a l r e s u l t s o f B.R. Lawn e t a l . /lo/. They u s e d r a l s o i n d e n t a t i o n f l a w s w i t h a l a r g e r a n g e o f i n d e n t a t i o n l o a d s (up t o 300 N). The measured r u p t u r e s t r e s s v a r i e d a s e x p e c t e d from f i g u r e 5.

As a c o n c l u s i o n s e v e r a l t r e n d s o b s e r v e d from e x p e r i m e n t a l r e s u l t s from t h e l i t e r a - t u r e c a n b e , a t l e a s t , q u a l i t a t i v e l y e x p l a i n e d u s i n g s i m u l a t i o n w i t h l o c a l e f f e c - t i v e c r a c k p r o p a g a t i o n and f r a c t u r e c r i t e r i a . I n t h e c a s e o f i n t r i n s i c o r n a t u r a l f l a w s , t h e number of p a r a m e t e r s u s e d i n t h e s i m u l a t i o n s h o u l d be e x t e n d e d and a d a p t e d t o e a c h t y p e o f f l a w . I n t h a t r e s p e c t t h e c h e m i c a l a s w e l l a s t h e morpholo- g i c a l v i c i n i t y of t h e f l a w must be t a k e n i n t o a c c o u n t i n o r d e r t o d e s c r i b e c o r r e c - t l y and a c c u r a t l y t h e f l a w growth p a t h .

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