Thermally Induced Residual Eutectic Composites

Stresses in

D. A. KOSS AND S. M. C O P L E Y

The effect of thermally induced residual stresses on the yield behavior of a unidirectionally solidified eutectic, (Co,Cr)-(Cr,CoI.~Cs, is presented. At low temperatures, the yield stress is found to depend on the sense of the applied stress. The difference in yield stress between ten- sion and compression is a function of temperature and disappears at a sufficiently high stress relaxation temperature. A straightforward analysis is presented that predicts the observed yielding behavior and a stress relaxation temperature that agrees well not only with the value obtained by observing the temperature dependence of the yield stress, but also with the value obtained from the thermal expansion behavior of the eutectic composite.

R E S I D U A L s t r e s s e s can d e v e l o p in a l i g n e d f i b e r c o m p o s i t e s for a v a r i e t y of r e a s o n s i n c l u d i n g t h e r m a l e x p a n s i o n m i s m a t c h , p h a s e t r a n s f o r m a t i o n s and n o n - u n i f o r m p l a s t i c f low. A l though the e f f e c t of s u c h s t r e s s e s on m e c h a n i c a l b e h a v i o r h a s b e e n r e c o g n i z e d by E b e r t e t a l . ~'3 and o t h e r s , 4-5 no w o r k to e v a l u a t e t h i s e f f e c t in a d i r e c t i o n a l l y s o l i d i f i e d e u t e c t i c c o m - p o s i t e h a s b e e n r e p o r t e d . In t h i s i n v e s t i g a t i o n , we e x a m i n e the e f f e c t of t h e r m a l l y i n d u c e d r e s i d u a l s t r e s s e s on the m e c h a n i c a l b e h a v i o r of t he ( C o , C r ) - (Cr ,Co) rC3 e u t e c t i c c o m p o s i t e .

R E S I D U A L T H E R M A L S T R E S S E S

T h i s s e c t i o n d e s c r i b e s the e f f e c t of t h e r m a l l y i n - d u c e d r e s i d u a l s t r e s s e s on the y i e l d b e h a v i o r of c o m - p o s i t e s . R e l a t i o n s a r e d e r i v e d w h i c h p r e d i c t t he m a g - n i t u d e of the r e s i d u a l s t r e s s in t e r m s of m e a s u r a b l e c o m p o s i t e p r o p e r t i e s .

A c c o r d i n g to L a s z l o , 8 the r e s i d u a l s t r e s s e s d e - v e l o p e d in a l a m e l l a r s l a b c o m p o s i t e due to t h e r m a l e x p a n s i o n m i s m a t c h a r e

0.R = V z M I M z 7 [ la]

R V 1 M l if'f2 72 V I M , + V z M z

Ei wlaere ,.l.l i = l - v i" T h e s e e q u a t i o n s a r e a p p r o x i m a t e b e -

c a u s e i t i s a s s u m e d t h a t s t r e s s e s d e v e l o p only in t he p l a n e of the l a m e l l a e , a r e c o n s t a n t a c r o s s the t h i c k - h e s s of a l a m e l l a , and a r e the s a m e for e v e r y d i r e c - t i on in the p l a n e . A s i m i l a r a n a l y s i s can b e a p p l i e d to a f i b e r c o m p o s i t e to show t h a t the a x i a l s t r e s s e s a r e a l s o g i v e n by E q s . [ l a ] and [ l b ] e x c e p t t h a t M i -= Ei . 7-~ In t h i s c a s e , it i s a s s u m e d t h a t s t r e s s e s d e v e l o p on ly

D. A. KOSS and S. M. COPLEY, formerly with the Advanced Mate- rials Research and Development Laboratory, Pratt & Whitney Aircraft. Middletown, Conn., are now Associate Professor, Department of Met- allu~cal Engineering, Michigan Technological University, tIoughton, Mich., and Associate Professor, Materials Science and Mechanical Engi- neering, University of Southern California, Los Angeles, Calif., rcspee- tivcly.

This paper is based on an invited talk presented at a symposium on Properties of Aligned Eutectic Composites sponsored by the tMD Com- posite Materials Committee held on October 13, 1969 at the Fall Meet- ing of The Metallur~cal Society. of AIME, in Philadelphia, Pa.

parallel to the fiber axis and are constant across each component of the composite.

Consider the case of a fiber composite cooling from the melt. We shall designate componen t 1 as the nzcL-

t r i x p h a s e , c o m p o n e n t 2 a s lhe f i b e r p h a s e , and a s - s u m e tha t c~i > ~2. Two r e g i m e s of b e h a v i o r c a n be identified:

1) For T > To, all stresses developed in the matrix and fibers upon cooling are relieved by creep, and

2) For T ~ T0, stresses develop in the matrix (ten- sile) and in the fibers (compressive), which increase linearly with decreasing temperature.

If a stress is applied to the composite parallel to the fiber direction, the resulting stresses in the ma- trix, fibers, and composite are related by the rule of mixtures '~

A A ~c = or, V~ + ~ V~ [2]

If t he f i b e r s and m a t r i x d e f o r m e l a s t i c a l l y , t h e n

A A 0"1 _ G2 E~ ~_~ [3]

T h e a p p l i e d s t r e s s in the m a t r i x i s o b t a i n e d by e l i m i -

n a t i n g o A f r o m E q s . [ 2 ] and [3] , w h i c h g i v e s

A ( e v3- ~ = V l + Z ~ y ec [4]

It i s a s s u m e d t h a t the f i b e r s a l w a y s r e m a i n e l a s t i c s o t h a t f o r T < T0, the p r o p o r t i o n a l l i m i t of the c o m p o s i t e i s r e a c h e d when the s u m of t h e a p p l i e d and r e s i d u a l s t r e s s e s in the m a t r L x b e c o m e s e q u a l to t he p r o p o r - t i o n a l l i m i t of the m a t r i x , i . e .

P

F o r t h i s r e g i m e , E q s . [ l a ] wi th M i = E i , [4] and [5] m a y b e c o m b i n e d to g ive t he p r o p o r t i o n a l l i m i t of the c o m p o s i t e

w h e r e the p lu s is fo r t e n s i o n a n d the m i n u s f o r c o m -

p r e s s i o n and cr P i s a l w a y s t a k e n to be p o s i t i v e . W i t h r e s i d u a l s t r e s s e s , t he p r o p o r t i o n a l l i m i t of the c o m - p o s i t e d e p e n d s on the s e n s e of the a p p l i e d un ia .x ia l s t r e s s . T h e d i f f e r e n c e b e t w e e n t he p r o p o r t i o n a l l i m i t

METALLURGICAL TRANSACTIONS VOLUME 2, JUNE 1971-1557

in compression and that in tension is

[ 7 t

which d e c r e a s e s with inc reas ing t e m p e r a t u r e becoming z e r o at the s t r e s s re laxa t ion t empe ra tu r e .

The s t r e s s re laxa t ion t e m p e r a t u r e can also be de- t e r m i n e d f rom the t h e r m a l expansion behavior of the compos i te . For T < To, the l inea r t he rma l expansion coef f ic ien t of the compos i te is given by u

7 V x E ~ x + ~. z E.~ctz

Fo r T > To, the m a t r i x c reeps and the t he rma l expan- sion coeff ic ient of the compos i te approaches that of the f iber . The change in slope m a r k s the s t r e s s r e l a x - ation t empera tu re , which provides an independent check of the theory.

The p reced ing ana lys i s can be extended into the p l a s t i c - e l a s t i c region with the r e s u l t that the d i f f e r - enee in flow s t r e s s between tension and c o m p r e s s i o n is s t i l l given by Eq. [ 7] and is independent of s t ra in . However , because of many effects that do depend on s t r a in , this ana lys i s is expected to be a good approx i - mat ion only at ve ry sma l l p las t ic s t ra ins .

The t he rma l ly induced r e s idua l s t r e s s ana lys i s p r e - sented he re modi f ies the rule of m i x t u r e s in a s t r a i g h t - fo rward manner . This ana lys i s will now be shown to d e s c r i b e adequately the yield behavior of the u n i d i r e c - t ional ly sol idif ied (Co,Cr) - (Cr ,Co)rCa eutec t ic . A m o r e detailed but complicated analysis, which takes into ac- count radial and tangential stresses, has been devel- oped by Ebert and coworkers. '-a

PROCEDURE

The (Co,Cr)- (Cr ,Co)7Ca eutect ic m a t e r i a l used in this inves t iga t ion was un id i rec t iona l ly sol idif ied f rom the m e l t in the fo rm of ~ and 1 in. rods . Rela t ive ly slow sol id i f ica t ion r a t e s were employed so that the cas t rods were , in effect , slow cooled f rom the me l t ing t e m p e r a t u r e

T e n s i l e and c o m p r e s s i o n spec imens were machined f rom the d i rec t iona l ly sol idif ied rods with the i r s t r e s s axis p a r a l l e l to the f iber axis . The tens i le spec imens had a gage sect ion ~ in. diam. by ~- in. long. The c o m - p r e s s i o n spec imens were r ec t angu la r pa ra l l e l ep ipeds 0.1 in. sq by 0.3 in. long or cy l indr ica l samples 0.125 in. d iam. and 0.320 in. long. The compres s ion data was the s a m e for both of these two spec imen shapes. The gage length of the tensile specimens and the whole of the compression specimens were electropolished in a solution of 10 pct. perchloric acid in methanol prior to testing.

All tensile testing was performed in a Centorr high vacuum testing chamber mounted on an h~stron univer- sal testing machine. The crosshead speed was adjusted to give a strain rate of 3 • 10-~sec -~. Test tempera- tures varied from room temperature to 1200~ and the temperature was controlled to ~:3~ All tensile tests were performed in vacuum at a pressure of approxi- mately 10 -~ Torr. Compression tests were performed in a direct push compression fixture in air. Since all compression tests were at temperatures where the eutectic alloy forms a protective chromium oxide scale, no adverse effects on the mechanical properties

�9 ,.. ; . ~ . . g : - : . . , . . < . .

" "":" . ~ . ' O .'0 ~ ~ .. ~ ' ~ ~ o " " . . .

..:. ~ - . ~ ..~'.," '~ :~.. .~ - : .-. " ,. r .q " i . 0

~,~.~,!,o:/:.:~!rk:~:i~. ~- :: <'i~';~; ::. ~:'</.A. V.....g,~~ ~ > ~ , ~ . :~:.', .:" ~.'~ i~ ".~ ~ ! , ~ . '~i ~ . : " " :~

" ..... ;9: ~ - ~:.i. " ~:" . . . . . . . . '~ . . . . "G~" " " ~ " :

�9 "~".s " ~x.~7~':.:... . -r ~ Z'~/.

~ & . / } ?

Fig. l--Microstructure of the unidirectionally solidified eu- tectic, (Co, Cr)-(Cr,Co)?C 3, with (a) transverse, and (b) lon- gitudinal sections shown.

would be expected dur ing s h o r t - t i m e c o m p r e s s i o n t es t s in a i r . All spec imens w e r e soaked for 0.5 hr. p r i o r to test ing to min imize e f fec ts due to var ia t ion in heat ing ra te .

The t he rm a l expansion behavior of s e v e r a l of the specimens was determined up to 900~'C employing a self-compensating quartz tube dilatometer.

RESULTS

Fig. 1 shows tile structure of the unidirectionally solidified (Co,Cr)-(Cr,Co)~Ca eutectic alloy both par- allel and transverse to the growth direction. It con- sists of aligned (Cr,Co}.~Cs carbides in a solid solu- tion matrix of Co-28 wt. pct. Cr. The chromium car- bides, which contain about 15 wt. pet. Co in solut ion, occupy about 30 volume pct. of the m a t e r i a l . It is e v i - dent f rom Fig. I that the ca rb ides a r e long in one d i - rec t ion and shor t in the two or thogonal d i r ec t ions and, t he re fo re , cau be r ega rded as f ibers . A de ta i led in- ves t igat ion of the morphology and growth characteris- tics of the (Co,Cr)-(Cr,Co)rC3 eutectic alloy has been reported by Thompson and Lemkey.'Z

Thompson el al. have obtained stress-strain curves in tension and compression for the (Co,Cr)-(Cr,Co)TC3

1558-VOLUM E 2, JUNE 1971 METALLURGICAL TRANSACTIONS

soo I 1 I I soo

2 5 0

2 0 0

150

I00

5 0

COMP CARBIDE REINFORCED

C O B A L T EUTECTIC

T=27 ~ C

ENSION

o I i I I 0.5 1.0 1.5 2.0 2.5

STRAIN (%}

Fig. 2--The stress-strain behavior of the (Co,Cr)-(Cr,Co)TC 3 eutectic tested in tension and compression {after Thompson et al.13).

eutect ic de fo rmed pa ra l l e l to the f iber axis at room t e m p e r a t u r e ; the i r r e su l t s a r e shown in Fig. 2. '~ The s t r e s s - s t r a i n cu rves can be d e s c r i b e d in t e r m s of c l a s s i c a l f iber compos i te behavior . At low s t r e s s e s both the f ibe rs m~d the m a t r i x deform e las t i ca l ly . The compos i te s t ra in is a l i nea r function of applied s t r e s s with a s lope de te rmined by the e l a s t i c constants of the m a t r i x and the f ibers . At higher s t r e s s e s the m a t r i x y ie lds and de fo rms p las t i ca l ly while the f ibe r s r emain e las t ic . The composi te s t ra in is again a l inear function of s t r e s s but with a new slope de t e rmined by the e l a s - t ic constants of the f ibe rs and the s t ra in hardening ra te of the ma t r ix . We shall r e f e r to the t rans i t ion point between these two r e g i m e s of behavior as the p ropor - t ional l imi t . It can be seen that at room t e m p e r a t u r e the p ropor t iona l l imi t in c o m p r e s s i o n exceeds that in tension by about 120,000 psi.

That the l a rge d i f fe rence in propor t iona l l imi t be- tween tension and c o m p r e s s i o n ex i s t s only in an aligned ahoy can be eas i ly demons t ra ted . Thompson e t a l . ~J

r e p o r t that a s low-coo led but nondi rec t iona l (Co,Cr)- (Cr,Co)7C~ eutect tc alloy showed a c o m p r e s s i v e yield s t r e s s of 145,000 psi. A s i m i l a r , s low-coo led but non- d i r ec t iona l sample was t es ted in tension and f rac tu red with no obse rvab le p las t ic s t ra in at 125,000 psi. It is r ea sonab le to conclude that the l a rge d i f fe rence be- tween c o m p r e s s i v e and t ens i l e yield s t r e s s e s occurs only when the eutect ic is in the fo rm of an al igned composi te .

The r e su l t s of al l the s t r e s s - s t r a i n m e a s u r e m e n t s obtained in this invest igat ion a re p re sen ted in Fig. 3. We have plot ted the 0.2 pct. y ie ld s t r e s s r a the r than the p ropor t iona l l imi t because it can be m o r e re l iab ly de t e rmined f rom s t r e s s - s t r a i n cu rves . The change in 0.2 pct. y ie ld s t r e s s between tension and compres s ion

I i I I I

,, \ COMPRESSION

a .

~1so o TENSION

, ~ P o o

o ! t I I I , I , 2 0 0 ~,00 6 0 0 6 0 0 1 0 0 0 1200

TEMPERATURE [ ~

Fig. 3--The temperature dependence of the yield stress for the (Co,Cr)-(Cr,Co)TC s eutectic as tested in compression and tension.

is approximately the same as the change in propor- tional limit, as would be expected.

DISCUSSION

Our analysis has indicated that thermally induced residual stresses can affect the stress-strain behavior of a eutectic composite. In particular, our analysis predicts that a difference in the proportional limit (or the 0.2 pct. yield stress) should be observed between tension and compression told that this difference should decrease monotonically with increasing temperature up to the stress relaxation temperature. Fig. 3 shows that the tensile and compressive yield stress do con- verge, as predicted, and that the stress relaxation temperature is between 700 ~ and 800~ A theoretical value for the stress relaxation temperature can be cal- culated by setting T = 70~ and A~ = 120,000 psi in Eq. [7] and solving for To. If we take ot~ = 15.0 • 10-B (C~) -I (measured), ctz = I0.0 • 10 -6(C~ -I (for pure Cr7C314), E~ = 30 • 106 psi and E2 = 43 x 108 psi, ~ a stress re- laxation temperature of 950~C is obtained. The agree- ment between predicted and observed results is rea- sonably good considering the sensitivity of To to the relative difference in thermal expansion coefficient. For example, if the measured composite thermal ex- pansion coefficient (~c = 12.2 )< 10 -6 (CO) ' ' ) and Eq. [8] are used to calculate Ac~ in Eq. [7], a lower relaxation temperature of 650~C is obtained. It should be recog- nized that the stress relaxation temperature is a func- tion of the cooling and heating rates; To increases with faster cooling rates. In this investigation, such effects were minimized by testing samples which were in a slow-cooled condition and were heated under similar conditions. Relaxation effects near To may exist, but these effects should be minor and would not affect the major conclusions of the study.

As discussed previously, an independent test of the magni tude of the s t r e s s re laxa t ion t e m p e r a t u r e can be

METALLURGICAL TRANSACTIONS VOLUME 2, JUNE 1971-1559

'Z

-r.

U

500

I I I

T- / /

' - / ~ / / / / I I I 6oo 700 eoo 900 TEMPERATURE (~

Fig. 4--The thermal expansion of the carbide-reinforced co- balt eutectic.

made by obse rv ing the t e m p e r a t u r e dependence of the t he rm a l expansion of the eutec t ic compos i te . At t e m - p e r a t u r e s below To, a compos i t e t he rma l expansion coeff ic ient given by Eq. [8] should be obse rved . At t e m p e r a t u r e s above T o , where yie ld ing occu r s in the m a t r i x , the t he rma l expansion r a t e should be that of the f iber . Fig. 4 shows t h e r m a l expansion data which indica tes that a chznge in s lope does in fact occur at t e m p e r a t u r e s cons i s ten t with the re laxa t ion t e m p e r a - lure p red ic ted f rom the yie ld s t r e s s data in Fig. 3. At t e m p e r a t u r e s below 700~ the t h e r m a l expansion co- ef f ic ient is 12.2 x 10 -6 (C~) - ' compared to a value of 12.5 • 10 -8 (C~ -t p red ic ted f rom Eq. [8]. At t e m p e r a - t u r e s above 800~ where the hop ~ fcc phase t r a n s i - tion occurs on heating, the t h e r m a l expansion r a t e c o m p a r e s favorably with that expected f rom the f iber ( a c = 10.5 x 10 -6 (C~)-~). The t r ans i t ion region between 700 r and 800~ is not fully unders tood but does indicate that the t rans i t ion f rom the r e l axed condition (T > 800~ to the fully cons t ra ined condition (T < 700~C) occu r s o v e r a range of t e m p e r a t u r e s . Thus, the t h e r m a l ex- pansion data indicates that s o m e re s idua l s t r e s s ex i s t s up to 800~ which c o m p a r e s favorably with To ~- 900~C p red ic t ed by Eq. [7], or with To = 700 ~ to 800"~C as ob- s e r v e d in Fig. 3.

CONCLUSION

A simplified analysis has been presented for the ef- fect of thermally induced residua/ stresses on the yield behavior of an eutectic composite. The analysis predicts a difference in yield stress between tension and compression that decreases monotonically with in- creasing temperature. Observations of the temperature dependence of the yield stress for the unidirectionally solidified eutectic, (Co,Cr)-(Cr,Co)TC3, show that com- pressive and tensile yield stresses do converge with increasing temperature. Given the difference in the

yield s t r e s s between tension and c o m p r e s s i o n at any given t e m p e r a t u r e , the ana lys i s p red ic t s a s t r e s s r e - la.xation t e m p e r a t u r e at which no re s idua l s t r e s s ex i s t s . The p red ic t ed re laxa t ion t e m p e r a t u r e a g r e e s wel l not only with that obtained by observ ing the t e m - p e r a t u r e dependence of the yield s t r e s s in tension and c o m p r e s s i o n but a lso with that obtained independently by observ ing the t h e r m a l expansion behavior of the ca rb ide r e in fo rced eutec t ic alloy.

AC KNOWLEDGMENTS

We would l ike to thank Dr. E. H. Thompson for his coopera t ion and sugges t ions and M e s s r s . E. L. Johnson and R. E. Doiron for e x p e r i m e n t a l a s s i s t ance .

LIST OF SYMBOLS

R cr i = r e s idua l s t r e s s in the components .

A cr~ = applied s t r e s s in the components .

a c = s t r e s s applied to the composi te .

V i = volume f rac t ion of the components .

E i = Young 's modulus of the components .

~i = P o i s s o n ' s ra t io of the components .

M i is defined in the text.

P P ~c ,a t = p ropor t iona l l imi t of the compos i te ith com-

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

A a = a . - a z , where a~ and a2 a r e the l inea r t he r - mal expansion coef f ic ien ts of components I and 2.

,~, T = T - T o , where T is the t e m p e r a t u r e and To is the s t r e s s re laxa t ion t e m p e r a t u r e .

REFERENCES

1. L. J. Ebert, C. H. Hamilton, and S. S. Hecker: AFML.TR-67-95 (April 1967) AD-814815.

2. L. J. Ebert, C. H. Hamilton, and S. S. Hecker: AFML-TR-68-71 (March 1968) AD-837237.

3. L. J. Ebert, R. J. Fedar, C. H. Hamilton, S. S. Hecker, and P. K. Wri#t : AFML-TR-69-129 (June 1969).

4. E. M. Lenoe: AI','ML-TR.67-125 (May 1967). 5. E. V. Summer: SAMPEJ.. 1966, vol. 10, F-I 1. 6. F. LasTJo: at. Iron Steel lnsL, 1943, vol. 148, pp. t 73-99. 7. V. V. Abramov: Residual Stresses ~md Strains in Metals. Mashgis, Moscow,

1963. 8. W. D. Kingrey: Introduction to Ceramics, p. 478, John Wiley, New York,

1960. 9. G. L. Denman: AFML-TR-65-279 (March 1966).

10. D. L. McDanets, R. W. Jech, and J. W. Weeton: MetalProg.. 1960, vo[. 78, pp. 118-21.

1 t. F. Laszlo: J. Iron Steellnst., I945, vol. 151, pp. 207-28. 12. E. R. Thompson and F. D. Lemky: Met. Traits., 1970, vol. 1, pp. 2799-2806. 13. E. R. Thompson, D. A. Koss, and J. C. Chesnutt: Met. l)'ans., 1970. vol. I, pp.

2807-13. 14. J. Hinnuber and O. Rudiger: Arch. Eisenhuettenw., 1953, vo]. 24, p. 267.

1 5 6 0 - V O L U M E "~ JUNE 1971 M E T A L L U R G I C A L TRANSA( .TIONS