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Quantitative fractography of creep cavitation
Scripta METALLURGrCA e r M A T E R~.~,~,A '" r
Voi.
24, pp. 1 3 3 5 - t 3 ~ ; 0 , t99~ Priated in t h e U . S . A .
P e r g a m o n P r e s s plc All rights reser--ed
QUANTITATIVE FRACTOGRAPHY OF ,CREEP CA.:ZT.-TZD: 3. Svoboda and V. S k l e n i ~ k a C z e c h o s l o v a k Academy of S c i e n c e s , i n s t i t u t e ot P h y s i c a l 616
62
Brno,
TM t allurgy ~,e
Czechoslovakia
{ . R e c e i v e d March ~9, 1990) ( R o ~ - i s e d Xlav ~ 1990)
Introduction H~gh t e m p e r a t u r e , low-ductility creep fracture i s g e n e r a l l y a c c e p t e d as a r e s u l t of t h e f o r m a t i o n and g r o w t h of i n t e r g r a n u l a r cavities. The q u a n t i f i c a t i o n of the three-dimensional cavitation structure has been most o f t e n p e r f o r m e d upon two-dimensional p l a n e s e c t i o n s of t h e c r e e p s p e c i m e n s by means of s t e r e o l o g i c a l procedures (1-3). I t does n o t a v o i d a p p r o x i m a t i o n s w h i c h can s e r v e u n t i l better solutions a r e f o u n d . However, t o be a c c e p t a o l e , a p p r o x i m a t i o n s s h o u l d be based upon c l e a r l y d e f i n e d g e o m e t r i c c o n c e p t s so t h a t t h e i r p r e c i s i o n can be a s s e s s e d . F u r t h e r m o r e , a number o f e x p e r i m e n t a l d i f f i c u l t i e s a r i s e from t h e s e c t i o n i n g and p o l i s h i n g t e c h n i q u e s , w h i c h may s i g n i f i c a n t l y influence t h e f e a t u r e s of cavities (4-5). Moreover, obtaining some i m p o r t a n t q u a n t i t a t i v e p a r a m e t e r s by means o f t w o - d i m e n s i o n a l sectioning i s e x t r e m e l y e l a b o r a t e and t i m e c o n s u m i n g . There exists a vast potential field for the application of q u a n t i t a t i v e m i c r o s c o p y to u n f l a t surfaces, for example, fracture s u r f a c e s ( 6 ) . The r e c e n t l y presented improved fractographic t e c h n i q u e s ( 7 , 8 ) make i t p o s s i b l e to p r o d u c e the artificial brittle intergranular fracture surface, preserving perfectly the microstructure of c a v i t i e s . Such a f r a c t u r e surface is convenient for the quantitative analysis. The p r e s e n t n o t e i s d e v o t e d to t h e method of q u a n t i f i c a t i o n of c r e e p c a v i t a t i o n damage upon t h e a r t i f i c i a l intergranular fracture. The m o d i f i e d l i q u i d metal embrittlement t e c h n i q u e was used to c r e a t e the a r t i f i c i a l intergranular brittle fracture s u r f a c e of p r e c r e p t n i c k e l s p e c i m e n s . The d e t a i l e d d e s c r i p t i o n of t h e f r a c t o g r a p h i c t e c h n i q u e and c r e e p t e s t c o n d i t i o n s were d e s c r i b e d e l s e where ( 8 ) . G r a i n b o u n d a r y d e t a i l s of interest have been e x a m i n e d w i t h a s c a n n i n g electron m i c r o s c o p e (SEM) by means of q u a n t i t a t i v e fractography. Fundamental Relationships The a p p r o a c h to t h e q u a n t i t a t i v e spatial reconstruction of t h e i n t e r n a l graphic viewpoint.
a n a l y s i s of c a v i t a t i o n p r o c e e d s from the fracture surface starting from a s t e r e o -
Let x be t h e a x i s of r o t a t i o n of t h e p r o j e c t i o n ( F i g . I ) , ~ . and ~2 t h e tilt a n g l e s w i t h r e s p e c t to t h e a x i s z ( i d e n t i c a l w i t h the t e n s i l e stress axis of t h e c r e e p s p e c i m e n ) , and ( x ~ , y l ) , ( x , yp) t h e c o o r d i n a t e s of t h e p o i n t s on the projections c o r r e s p o n d i n g @o t ~ e p o i n t w i t h t h e c o o r d i n a t e s ( x , y , z) o f t h e r e f e r e n c e s y s t e m c h o s e n . Then X = x!
(1)
= x2,
y = (Y2sin/01
-
z
= (YlCOS~2
- Y2COS/~l)/D
O
= sin~lcos~2
(2)
YlSin'g2)/D, ,
(3)
sin~2cos~l.
(a)
where -
1335 0036-q748/qP $ 3 . 0 0 + .00 Copyright ( c ) 1990 P e r g a m o a P r e s s -
t
-
o
plc
1556
CREEP CAVITATION
l Z
Vo[.
The b e s t r e s u l t s Z - #~ £ (30c'
"12_
2~,
can be o b t a i n e d 50°)"
No,
7
for
To e v a l u a t e t h e f r a c t u r e surface f e a t u r e s from t h e two s t e r e o m i c r o graphs, the perimeters of cavities have been d i g i t i z e d by u s i n g a se~ o f p o i n t s ( f r o m 20 to 5 0 ) . Hence, the c a v i t y p e r i m e t e r s have been a p p r o x i m a t e d by p o l y g o n s . The o b t a i n e d r e s u l t s show t h a t the just mentioned approximation is quite acceptable. When t h e s [ e r e o m i c r o X=XI=X2 ~ -~ g r a p h s a r e p r o d u c e d i n SEN, the s p e c i m e n has to be t i l t e d ; t h i s may cause a slight shift o f t h e s p e c i m e n i n the 0 A x m i c r o s c o p e and a s l i g h t change i n t h e magnification o f the second m i c r o g r a p h . FIG. 1 The n e c e s s a r y f i t t i n g of t h e s t e r e o Reconstruction o f space c o o r d i n a t e s of m i c r o g r a p h s has been c a l c u l a t e d utilizp o i n t X from c o o r d i n a t e s of its proing relation ( I ) as t h e c o n d i t i o n of t h e minimum sum o f s q u a r e s o f x - c o o r d i jections X 1 and X2, nates differences of g r a v i t y centers of corresponding cavities in the stereomicrographs. After the fitting the x-coordinates differences were l o w e r t h a n the resolution o f t h e m i c r o g r a p h s . The r e c o n s t r u c t i o n of c a v i t y p e r i m e t e r s i n three-dimens±onal space has been p e r f o r m e d i n such a way t h a t , for a given value of x , c o r r e s p o n d i n g c o o r d i n a t e s y~ and y on p r o j e c t i o n s of cavity perimeter were c a l c u l a t e d ; then the coordinates in2three-dimensional space were e v a l u a t e d u s i n g t h e e q u a t i o n s ( 1 ) to ( 4 ) . As t h e r e c o n s t r u c t e d c a v i t y p e r ± m e t e r s were s e r r a t e d due t o i n a c c u r a c i e s at the cavity perimeters digitalization in stereom i o r o g r a p h s and t h e i r a p p r o x i m a t i o n by p o l y g o n s , t h e r e c o n s t r u c t e d cavity perim e t e r s had t o be s m o o t h e d . T h a t i s why t h e r e c o n s t r u c t e d p o i n t s of each c a v i t y p e r i m e t e r were f i t t e d to t h e b i c u b i c p o l y n o m i a l z ( x , y) and t h e smoothed c a v i t y p e r i m e t e r was o b t a i n e d as t h e p r o j e c t i o n of the cavity perimeter in the first stereomicrograph into the surface z (x, y). With the aid of two-dimensional splines, t h e smoothed c a v i t y p e r i m e t e r s were l i n k e d by a s u r f a c e a p p r o x i m a t i n g t h e g r a i n b o u n d a r y , w h i c h was c a l c u l a t e d in a rectangular equidistant g r i d o f 51x51 p o i n t s .
/
~
Results The e f f e c t i v e n e s s o f t h e p r o p o s e d p r o c e d u r e i s d e m o n s t r a t e d on an e x a m p l e of t h e r e c o n s t r u c t i o n of the internal fracture s u r f a c e (LME T e c h n i q u e ) o f n i c k e l a f t e r h i g h t e m p e r a t u r e c r e e p . From t h e same p l a c e o f t h i s f r a c t u r e surf a c e two s t e r e o m i c r o g r a p h s were t a k e n ( F i g s . 2a and 2b) s h £ w i n g t h e p r e s e n c e o f intergranular creep cavities. The t i l t a n g l e s a r e ~ I = 1 0 - ( F i g . 2a) and ~ o = = 45 ° ( F i g . 2 b ) . The r e s u l t i n g reconstruction of cavitated grain boundary facet in parallel projection i s shown i n F i g . 3. The t o p o l o g y o f t h e same g r a i n boundary facet in the isohypse description in the plane fitting the recons t r u c t e d c a v i t y p e r i m e t e r s and i n t h e p l a n e n o r m a l t o z a x i s a r e p l o t t e d in F i g . 4 and F i g . 5, r e s p e c t i v e l y . A more c o m p l i c a t e d r e c o n s t r u c t i o n o f two a d joining g r a i n b o u n d a r y f a c e t s from F i g . 6 ( t h e b r i g h t n e s s is differentiated in the horizontal direction and n e g a t i v e i n o r d e r t o i m p r o v e t h e s h a r p n e s s of c a v ity perimeters for digitalization) is in Fig. 7 (isohypse description in the plane normal to z axis). Each i s o h y p s e r e p r e s e n t s 2 #m. The n u m e r i c a l d a t a o b t a i n e d f r o m t h e s t e r e o m i c r o g r a p h s reconstruction (Fig. 2 co F i g . 5) make i t p o s s i b l e to e v a l u a t e a number o f i m p o r t a n t p a r a m e t e r s on cavitation damage. One o f t h e most i m p o r t a n t q u a n t i t a t i v e characteristics of a c a v i t y i s t h e g r a i n b o u n d a r y a r e a Sc o c c u p i e d by t h e c a v i t y , w h i c h can be c a l c u lated as:
Sc "
Sp
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)'
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Vol
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No.
7
CRE=_P
. . . . . . . . . ~N
133
i
L (a
FIG.
2.
(b)
Stereomicrographs
~
of
internal
~
~
3.
surface
7
,
FIG.
fracture
~
of
crept
nickel.
_- ~ /
~
i
Reconstruction of cavitated F i g s . 2a and 2b tn p a r a l l e l
grain boundary projection.
facet
from
where S i s the c a v i t y area i n a p r o j e c t i o n , iJ i s the a n g l e between t h e p r o j e c t ion direction and n o r m a l t o t h e e l e m e n t dS, and < L / c o s ~ > is the average value of $/cos~ Ln t h e g r a i n b o u n d a r y area o c c u p i e d by the c a v i t y . If the grain b o u n d a r y s u r f a c e i s d e s c r i b e d by t h e v a l u e s of z ± , j = z ( x i , yj)±,j = C, L . . . . , , , 50 r e p r e s e n t i n g the value of the rectangular equidistant grid direction,
coe~
then
1+
the value
J +~'T
of
z c o o r d i n a t e o f the g r a i n x i , y j and z d i r e c t i o n is
~ in
=
I+
the position
x±,
+l'J-zi-i'J 2&x
yj
-+'
boundary e~rface the projection
can be c a l c u l a t e d
in
from
i'J +1-~i ",
2&y
/~
w h e r e A x and A y a r e t h e g r i d - p o i n t d i s t a n c e s i n t h e x and y d i r e c t i o n s . The value can be e s t i m a t e d as t h e a r i t h m e t i c a v e r a g e of v a l u e s of Z / c o s ' ~ in the grid points Lying in the cavity, The a r e a S of z a x i s p r o j e c t i o n o f cav P c c c be c a tcui t y t h e p e r i m e t e r o f w h i c h i s d e s c r i b e d by p o i n t s (xi' Y i ' z i ) can lated Figs,
as t h e a r e a
of
a poZygon w i t h
apices
( x c,_ y C ) .
The q u a n t i t a t i v e characteristics of c a v i t i e s 2a and 2b a r e s u m m a r i z e d i n T a b l e 1.
from
stereomicrograohs
in
1338
CREEP CAVITATION
t!~';.;'.\ ~ - - ~
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/ ,, i
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:1
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FIG. 4
T o p o l o g y o f g r a i n b o u n d a r y f a c e t from F i g s , 2a and 2b i n t h e i s o h y p s e d e s c r i p t i o n in the plane fitting the reconstructed cavity perimeters.
I
P
lO#m FIG.
1
5
T o p o l o g y of g r a i n b o u n d a r y f a c e t from F i g s . 2a and 2b i n the isohypse description in the plane normal to the tensile stress axis.
24, No.
7
Voi
~,
,:~
CREEP C kV[TAT ~
FIG. 6 One crept
of the s t e r e o m i c r o g r a p h s of i n t e r n a l f r a c t u r e s u r f a c e of n±ckel; the b r i g h t n e s s is d i f f e r e n [ ± a t e d in [he h o r i z o n t a l d ± r e c t i o n and negative. .<
r-1
o~ 28-
°z
\
Ax~s~ \\
/~xs
•
< i:i#
0 !
I
-sb-4-~0 6 20 40 s'o TrLT ANOLE OF PROJECT,0N 100 ~Jm
FIG. 7 Reconstruction Fig. 6 in the plane normal
of two adjoinZng facets from isohypse descript&on in the to the tensile stress ax±s.
PLANE [ DEG] FIG.
8
Dependence of the projected cavity area fraction on t h e tilt angle of p r o j e c t i o n plane
CREEP CAVITATION
1340
Vol.
24, No. 7
TABLE ! Cavity
No.
1 2 3 4 5 6 7 8 9 I0 11 12 13 14 15 16 17 18 19 20 The q u a n t i t a t i v e are as follows:
,/ y 2 [ °] 54,24 53.17 48.39 49.$0 43.49 52.73 41.78 43.24 56.63 25.98 56.99 29.07 42,01 51.62 66.25 62.81 45.34 48.11 44.88 26.59 characteristics
1 . 715 1.670 1.550 1.529 1.386 1,681 1,359 1,374 1,845 1,133 1,843 1.148 1.454 1.620 2.505 2.220 1.430 1.529 1.508 1.120 of
[ h-I "m ? 21j Sp~aV
6.152 2.228 4,756 19,800 21.467 22.636 13.918 6.840 7.113 20.856 5.875 6.625 2.938 4.0!6 0,827 3.282 1.326 2.557 10.018 i0,000
the whole g r a i n
scc[I0-12m2 ] 10.556 3.723 7.376 30.286 29.774 38.072 18.915 9.403 13.129 23.642 10.851 7.609 4.274 6.508 2.074 7.289 1.898 3.911 15.115 11.208
boundary facet
P r o j e c t e d area of g r a i n boundary f a c e t a n a l y s e d True surface area of grain boundary facet analysed P r o 3 e c t e d a r e a o f g r a i n b o u n d a r y u n o c c u p i e d by c a v i t i e s T r u e s u r f a c e a r e a o¢ g r a i n b o u n d a r y u n o c c u p i e d by c a v i t i e s Area fraction of cavities in the projection True area fraction of c a v i t i e s <~> o f t h e g r a i n b o u n d a r y f a c e t a n a l y s e d < ~ > o f t h e g r a i n b o u n d a r y a r e a u n o c c u p i e d by c a v i t i e s
=
= = = =
analysed
646.25x10-~ 1013.77x10"{~ 473.01x10"~ 7 5 8 . 1 7 x i 0 -1 0.2681 0.252~ 47.77049.152 ° .
2 m2 m? m~ m"
One o f t h e aims o f the p a p e r i s t o d e t e r m i n e how t h e t r u e a r e a f r a c t i o n of c a v i t i e s may d i f f e r from t h e a r e a f r a c t i o n of cavities o b t a i n e d from t h e projection. T h a t i s why t h e r e c o n s t r u c t e d g r a i n b o u n d a r y s u r f a c e has been p r o jected into various directions. The r e s u l t s a r e p l o t t e d i n F i g . 8. I t can be c o n c l u d e d t h a t i n t h i s case t h e c a v i t y a r e a f r a c t i o n d e t e r m i n e d from t h e p r o 3 a c t i o n e s t i m a t e s the t r u e c a v i t y area f r a c t i o n with the relative accuracy w i t h i n 10 %, The e s t i m a t i o n o f t h e c a v i t y a r e a f r a c t i o n from p r o j e c t i o n s is the better, t h e more p l a n a r t h e g r a i n b o u n d a r y f a c e t . Note The main p r o b l e m i n h i b i t i n g an e a s y and q u i c k a p p l i c a t i o n of the p r o c e d u r e s p r e s e n t e d is the problem of a f a s t d i g i t a l i z a t i o n of cavity perimeters in ster e o m i c r o g r a p h s . The a u t h o r s b e l i e v e t h a t t h e r a p i d l y d e v e l o p i n g s e m i a u t o m a t i c and a u t o m a t i c image a n a l y s i s p r o c e d u r e s may m a s t e r t h i s p r o b l e m v e r y s o o n . References 1. 2. 3. 4. 5. 6. 7. 8.
E.E.Underwood, Quantitative S t e r e o l o g y , R e a d i n g , A d d i s o n - W e s l e y P u b l . Comp,, Massachusetts (1970), R . T . D e H o f f , M e t a l s Forum 5, 4 ( 1 9 8 2 ) . I.Saxl, V . S k l e n i 6 k a and 3 . ~ a d e k , Z s . M e t a l l k d e 72, 499 ( 1 9 8 1 ) . I<.Proch~zka, I.Saxl, V . S k l e n i 6 k a and 3 . ~ a d e k , S c r . M e t a l l . 17, 779 ( ! 9 8 3 ) . D , M . R . T a p l i n and O . L . D u n l o p , M e t a l s Forum 4, 69 ( 1 9 8 ! ) . M . C o s t e r and 3 . L , C h e r m a n t , I n t . M e t a l s Ray, 28, 228 ( 1 9 8 3 ) , T.C,Reiley, Scr. Metall. 15, 497 ( 1 9 8 1 ) . 3 . S v o b o d a and V . S k l e n i ~ k a , S c r . M e t a l l , , in press.
Voi.
24, pp. 1 3 3 5 - t 3 ~ ; 0 , t99~ Priated in t h e U . S . A .
P e r g a m o n P r e s s plc All rights reser--ed
QUANTITATIVE FRACTOGRAPHY OF ,CREEP CA.:ZT.-TZD: 3. Svoboda and V. S k l e n i ~ k a C z e c h o s l o v a k Academy of S c i e n c e s , i n s t i t u t e ot P h y s i c a l 616
62
Brno,
TM t allurgy ~,e
Czechoslovakia
{ . R e c e i v e d March ~9, 1990) ( R o ~ - i s e d Xlav ~ 1990)
Introduction H~gh t e m p e r a t u r e , low-ductility creep fracture i s g e n e r a l l y a c c e p t e d as a r e s u l t of t h e f o r m a t i o n and g r o w t h of i n t e r g r a n u l a r cavities. The q u a n t i f i c a t i o n of the three-dimensional cavitation structure has been most o f t e n p e r f o r m e d upon two-dimensional p l a n e s e c t i o n s of t h e c r e e p s p e c i m e n s by means of s t e r e o l o g i c a l procedures (1-3). I t does n o t a v o i d a p p r o x i m a t i o n s w h i c h can s e r v e u n t i l better solutions a r e f o u n d . However, t o be a c c e p t a o l e , a p p r o x i m a t i o n s s h o u l d be based upon c l e a r l y d e f i n e d g e o m e t r i c c o n c e p t s so t h a t t h e i r p r e c i s i o n can be a s s e s s e d . F u r t h e r m o r e , a number o f e x p e r i m e n t a l d i f f i c u l t i e s a r i s e from t h e s e c t i o n i n g and p o l i s h i n g t e c h n i q u e s , w h i c h may s i g n i f i c a n t l y influence t h e f e a t u r e s of cavities (4-5). Moreover, obtaining some i m p o r t a n t q u a n t i t a t i v e p a r a m e t e r s by means o f t w o - d i m e n s i o n a l sectioning i s e x t r e m e l y e l a b o r a t e and t i m e c o n s u m i n g . There exists a vast potential field for the application of q u a n t i t a t i v e m i c r o s c o p y to u n f l a t surfaces, for example, fracture s u r f a c e s ( 6 ) . The r e c e n t l y presented improved fractographic t e c h n i q u e s ( 7 , 8 ) make i t p o s s i b l e to p r o d u c e the artificial brittle intergranular fracture surface, preserving perfectly the microstructure of c a v i t i e s . Such a f r a c t u r e surface is convenient for the quantitative analysis. The p r e s e n t n o t e i s d e v o t e d to t h e method of q u a n t i f i c a t i o n of c r e e p c a v i t a t i o n damage upon t h e a r t i f i c i a l intergranular fracture. The m o d i f i e d l i q u i d metal embrittlement t e c h n i q u e was used to c r e a t e the a r t i f i c i a l intergranular brittle fracture s u r f a c e of p r e c r e p t n i c k e l s p e c i m e n s . The d e t a i l e d d e s c r i p t i o n of t h e f r a c t o g r a p h i c t e c h n i q u e and c r e e p t e s t c o n d i t i o n s were d e s c r i b e d e l s e where ( 8 ) . G r a i n b o u n d a r y d e t a i l s of interest have been e x a m i n e d w i t h a s c a n n i n g electron m i c r o s c o p e (SEM) by means of q u a n t i t a t i v e fractography. Fundamental Relationships The a p p r o a c h to t h e q u a n t i t a t i v e spatial reconstruction of t h e i n t e r n a l graphic viewpoint.
a n a l y s i s of c a v i t a t i o n p r o c e e d s from the fracture surface starting from a s t e r e o -
Let x be t h e a x i s of r o t a t i o n of t h e p r o j e c t i o n ( F i g . I ) , ~ . and ~2 t h e tilt a n g l e s w i t h r e s p e c t to t h e a x i s z ( i d e n t i c a l w i t h the t e n s i l e stress axis of t h e c r e e p s p e c i m e n ) , and ( x ~ , y l ) , ( x , yp) t h e c o o r d i n a t e s of t h e p o i n t s on the projections c o r r e s p o n d i n g @o t ~ e p o i n t w i t h t h e c o o r d i n a t e s ( x , y , z) o f t h e r e f e r e n c e s y s t e m c h o s e n . Then X = x!
(1)
= x2,
y = (Y2sin/01
-
z
= (YlCOS~2
- Y2COS/~l)/D
O
= sin~lcos~2
(2)
YlSin'g2)/D, ,
(3)
sin~2cos~l.
(a)
where -
1335 0036-q748/qP $ 3 . 0 0 + .00 Copyright ( c ) 1990 P e r g a m o a P r e s s -
t
-
o
plc
1556
CREEP CAVITATION
l Z
Vo[.
The b e s t r e s u l t s Z - #~ £ (30c'
"12_
2~,
can be o b t a i n e d 50°)"
No,
7
for
To e v a l u a t e t h e f r a c t u r e surface f e a t u r e s from t h e two s t e r e o m i c r o graphs, the perimeters of cavities have been d i g i t i z e d by u s i n g a se~ o f p o i n t s ( f r o m 20 to 5 0 ) . Hence, the c a v i t y p e r i m e t e r s have been a p p r o x i m a t e d by p o l y g o n s . The o b t a i n e d r e s u l t s show t h a t the just mentioned approximation is quite acceptable. When t h e s [ e r e o m i c r o X=XI=X2 ~ -~ g r a p h s a r e p r o d u c e d i n SEN, the s p e c i m e n has to be t i l t e d ; t h i s may cause a slight shift o f t h e s p e c i m e n i n the 0 A x m i c r o s c o p e and a s l i g h t change i n t h e magnification o f the second m i c r o g r a p h . FIG. 1 The n e c e s s a r y f i t t i n g of t h e s t e r e o Reconstruction o f space c o o r d i n a t e s of m i c r o g r a p h s has been c a l c u l a t e d utilizp o i n t X from c o o r d i n a t e s of its proing relation ( I ) as t h e c o n d i t i o n of t h e minimum sum o f s q u a r e s o f x - c o o r d i jections X 1 and X2, nates differences of g r a v i t y centers of corresponding cavities in the stereomicrographs. After the fitting the x-coordinates differences were l o w e r t h a n the resolution o f t h e m i c r o g r a p h s . The r e c o n s t r u c t i o n of c a v i t y p e r i m e t e r s i n three-dimens±onal space has been p e r f o r m e d i n such a way t h a t , for a given value of x , c o r r e s p o n d i n g c o o r d i n a t e s y~ and y on p r o j e c t i o n s of cavity perimeter were c a l c u l a t e d ; then the coordinates in2three-dimensional space were e v a l u a t e d u s i n g t h e e q u a t i o n s ( 1 ) to ( 4 ) . As t h e r e c o n s t r u c t e d c a v i t y p e r ± m e t e r s were s e r r a t e d due t o i n a c c u r a c i e s at the cavity perimeters digitalization in stereom i o r o g r a p h s and t h e i r a p p r o x i m a t i o n by p o l y g o n s , t h e r e c o n s t r u c t e d cavity perim e t e r s had t o be s m o o t h e d . T h a t i s why t h e r e c o n s t r u c t e d p o i n t s of each c a v i t y p e r i m e t e r were f i t t e d to t h e b i c u b i c p o l y n o m i a l z ( x , y) and t h e smoothed c a v i t y p e r i m e t e r was o b t a i n e d as t h e p r o j e c t i o n of the cavity perimeter in the first stereomicrograph into the surface z (x, y). With the aid of two-dimensional splines, t h e smoothed c a v i t y p e r i m e t e r s were l i n k e d by a s u r f a c e a p p r o x i m a t i n g t h e g r a i n b o u n d a r y , w h i c h was c a l c u l a t e d in a rectangular equidistant g r i d o f 51x51 p o i n t s .
/
~
Results The e f f e c t i v e n e s s o f t h e p r o p o s e d p r o c e d u r e i s d e m o n s t r a t e d on an e x a m p l e of t h e r e c o n s t r u c t i o n of the internal fracture s u r f a c e (LME T e c h n i q u e ) o f n i c k e l a f t e r h i g h t e m p e r a t u r e c r e e p . From t h e same p l a c e o f t h i s f r a c t u r e surf a c e two s t e r e o m i c r o g r a p h s were t a k e n ( F i g s . 2a and 2b) s h £ w i n g t h e p r e s e n c e o f intergranular creep cavities. The t i l t a n g l e s a r e ~ I = 1 0 - ( F i g . 2a) and ~ o = = 45 ° ( F i g . 2 b ) . The r e s u l t i n g reconstruction of cavitated grain boundary facet in parallel projection i s shown i n F i g . 3. The t o p o l o g y o f t h e same g r a i n boundary facet in the isohypse description in the plane fitting the recons t r u c t e d c a v i t y p e r i m e t e r s and i n t h e p l a n e n o r m a l t o z a x i s a r e p l o t t e d in F i g . 4 and F i g . 5, r e s p e c t i v e l y . A more c o m p l i c a t e d r e c o n s t r u c t i o n o f two a d joining g r a i n b o u n d a r y f a c e t s from F i g . 6 ( t h e b r i g h t n e s s is differentiated in the horizontal direction and n e g a t i v e i n o r d e r t o i m p r o v e t h e s h a r p n e s s of c a v ity perimeters for digitalization) is in Fig. 7 (isohypse description in the plane normal to z axis). Each i s o h y p s e r e p r e s e n t s 2 #m. The n u m e r i c a l d a t a o b t a i n e d f r o m t h e s t e r e o m i c r o g r a p h s reconstruction (Fig. 2 co F i g . 5) make i t p o s s i b l e to e v a l u a t e a number o f i m p o r t a n t p a r a m e t e r s on cavitation damage. One o f t h e most i m p o r t a n t q u a n t i t a t i v e characteristics of a c a v i t y i s t h e g r a i n b o u n d a r y a r e a Sc o c c u p i e d by t h e c a v i t y , w h i c h can be c a l c u lated as:
Sc "
Sp
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)'
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No.
7
CRE=_P
. . . . . . . . . ~N
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Stereomicrographs
~
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~
~
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7
,
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~
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Reconstruction of cavitated F i g s . 2a and 2b tn p a r a l l e l
grain boundary projection.
facet
from
where S i s the c a v i t y area i n a p r o j e c t i o n , iJ i s the a n g l e between t h e p r o j e c t ion direction and n o r m a l t o t h e e l e m e n t dS, and < L / c o s ~ > is the average value of $/cos~ Ln t h e g r a i n b o u n d a r y area o c c u p i e d by the c a v i t y . If the grain b o u n d a r y s u r f a c e i s d e s c r i b e d by t h e v a l u e s of z ± , j = z ( x i , yj)±,j = C, L . . . . , , , 50 r e p r e s e n t i n g the value of the rectangular equidistant grid direction,
coe~
then
1+
the value
J +~'T
of
z c o o r d i n a t e o f the g r a i n x i , y j and z d i r e c t i o n is
~ in
=
I+
the position
x±,
+l'J-zi-i'J 2&x
yj
-+'
boundary e~rface the projection
can be c a l c u l a t e d
in
from
i'J +1-~i ",
2&y
/~
w h e r e A x and A y a r e t h e g r i d - p o i n t d i s t a n c e s i n t h e x and y d i r e c t i o n s . The value
as t h e a r e a
of
a poZygon w i t h
apices
( x c,_ y C ) .
The q u a n t i t a t i v e characteristics of c a v i t i e s 2a and 2b a r e s u m m a r i z e d i n T a b l e 1.
from
stereomicrograohs
in
1338
CREEP CAVITATION
t!~';.;'.\ ~ - - ~
/
?
/ /
,-. . . . d , \
/ ,, i
',
; ,
'i
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Vol.
\------..
'
,"'.,
:1
:\"
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FIG. 4
T o p o l o g y o f g r a i n b o u n d a r y f a c e t from F i g s , 2a and 2b i n t h e i s o h y p s e d e s c r i p t i o n in the plane fitting the reconstructed cavity perimeters.
I
P
lO#m FIG.
1
5
T o p o l o g y of g r a i n b o u n d a r y f a c e t from F i g s . 2a and 2b i n the isohypse description in the plane normal to the tensile stress axis.
24, No.
7
Voi
~,
,:~
CREEP C kV[TAT ~
FIG. 6 One crept
of the s t e r e o m i c r o g r a p h s of i n t e r n a l f r a c t u r e s u r f a c e of n±ckel; the b r i g h t n e s s is d i f f e r e n [ ± a t e d in [he h o r i z o n t a l d ± r e c t i o n and negative. .<
r-1
o~ 28-
°z
\
Ax~s~ \\
/~xs
•
< i:i#
0 !
I
-sb-4-~0 6 20 40 s'o TrLT ANOLE OF PROJECT,0N 100 ~Jm
FIG. 7 Reconstruction Fig. 6 in the plane normal
of two adjoinZng facets from isohypse descript&on in the to the tensile stress ax±s.
PLANE [ DEG] FIG.
8
Dependence of the projected cavity area fraction on t h e tilt angle of p r o j e c t i o n plane
CREEP CAVITATION
1340
Vol.
24, No. 7
TABLE ! Cavity
No.
1 2 3 4 5 6 7 8 9 I0 11 12 13 14 15 16 17 18 19 20 The q u a n t i t a t i v e are as follows:
,/ y 2 [ °] 54,24 53.17 48.39 49.$0 43.49 52.73 41.78 43.24 56.63 25.98 56.99 29.07 42,01 51.62 66.25 62.81 45.34 48.11 44.88 26.59 characteristics
[ h-I "m ? 21j Sp~aV
6.152 2.228 4,756 19,800 21.467 22.636 13.918 6.840 7.113 20.856 5.875 6.625 2.938 4.0!6 0,827 3.282 1.326 2.557 10.018 i0,000
the whole g r a i n
scc[I0-12m2 ] 10.556 3.723 7.376 30.286 29.774 38.072 18.915 9.403 13.129 23.642 10.851 7.609 4.274 6.508 2.074 7.289 1.898 3.911 15.115 11.208
boundary facet
P r o j e c t e d area of g r a i n boundary f a c e t a n a l y s e d True surface area of grain boundary facet analysed P r o 3 e c t e d a r e a o f g r a i n b o u n d a r y u n o c c u p i e d by c a v i t i e s T r u e s u r f a c e a r e a o¢ g r a i n b o u n d a r y u n o c c u p i e d by c a v i t i e s Area fraction of cavities in the projection True area fraction of c a v i t i e s <~> o f t h e g r a i n b o u n d a r y f a c e t a n a l y s e d < ~ > o f t h e g r a i n b o u n d a r y a r e a u n o c c u p i e d by c a v i t i e s
=
= = = =
analysed
646.25x10-~ 1013.77x10"{~ 473.01x10"~ 7 5 8 . 1 7 x i 0 -1 0.2681 0.252~ 47.77049.152 ° .
2 m2 m? m~ m"
One o f t h e aims o f the p a p e r i s t o d e t e r m i n e how t h e t r u e a r e a f r a c t i o n of c a v i t i e s may d i f f e r from t h e a r e a f r a c t i o n of cavities o b t a i n e d from t h e projection. T h a t i s why t h e r e c o n s t r u c t e d g r a i n b o u n d a r y s u r f a c e has been p r o jected into various directions. The r e s u l t s a r e p l o t t e d i n F i g . 8. I t can be c o n c l u d e d t h a t i n t h i s case t h e c a v i t y a r e a f r a c t i o n d e t e r m i n e d from t h e p r o 3 a c t i o n e s t i m a t e s the t r u e c a v i t y area f r a c t i o n with the relative accuracy w i t h i n 10 %, The e s t i m a t i o n o f t h e c a v i t y a r e a f r a c t i o n from p r o j e c t i o n s is the better, t h e more p l a n a r t h e g r a i n b o u n d a r y f a c e t . Note The main p r o b l e m i n h i b i t i n g an e a s y and q u i c k a p p l i c a t i o n of the p r o c e d u r e s p r e s e n t e d is the problem of a f a s t d i g i t a l i z a t i o n of cavity perimeters in ster e o m i c r o g r a p h s . The a u t h o r s b e l i e v e t h a t t h e r a p i d l y d e v e l o p i n g s e m i a u t o m a t i c and a u t o m a t i c image a n a l y s i s p r o c e d u r e s may m a s t e r t h i s p r o b l e m v e r y s o o n . References 1. 2. 3. 4. 5. 6. 7. 8.
E.E.Underwood, Quantitative S t e r e o l o g y , R e a d i n g , A d d i s o n - W e s l e y P u b l . Comp,, Massachusetts (1970), R . T . D e H o f f , M e t a l s Forum 5, 4 ( 1 9 8 2 ) . I.Saxl, V . S k l e n i 6 k a and 3 . ~ a d e k , Z s . M e t a l l k d e 72, 499 ( 1 9 8 1 ) . I<.Proch~zka, I.Saxl, V . S k l e n i 6 k a and 3 . ~ a d e k , S c r . M e t a l l . 17, 779 ( ! 9 8 3 ) . D , M . R . T a p l i n and O . L . D u n l o p , M e t a l s Forum 4, 69 ( 1 9 8 ! ) . M . C o s t e r and 3 . L , C h e r m a n t , I n t . M e t a l s Ray, 28, 228 ( 1 9 8 3 ) , T.C,Reiley, Scr. Metall. 15, 497 ( 1 9 8 1 ) . 3 . S v o b o d a and V . S k l e n i ~ k a , S c r . M e t a l l , , in press.