The effects of physical parameters on plastic deformation instability

Scripta METALLURGICA

Vol. 12, pp. 1083-1086, 197S Printed in the United States

Pergamon Press, Inc

THE E F F E C T S OF P H Y S I C A L P A R A M E T E R S O N P L A S T I C D E F O R M A T I O N I N S T A B I L I T Y

G. MALAPRADE, D. ROUBY, G. F A N T O Z Z I G r o u p e d ' E t u d e s de M ~ t a l l u r g i e P h y s i q u e et de P h y s i q u e des M a t 4 r i a u x E.R.A. n°463 - I.N.S.A. de L Y O N - B ~ t i m e n t 502 - 20, avenue A. E i n s t e i n 69621VILLEURBANNE CEDEX - FRANCE

(Received July 20, 1973) (Revised October 5, 1978)

The s o - c a l l e d p l a s t i c instability, o b s e r v e d d u r i n g tensile t e s t i n g m e a n s a p r o g r e s s i v e d e c r e a s e of the a p p l i e d force, w h i c h i n d i c a t e s a d e f o r m a t i o n of the s p e c i m e n in a v e r y l o c a l i z e d area, l e a d i n g to necking. This d e f o r m a t i o n i n s t a b i l i t y u s u a l l y appears at low t e m p e r a t u r e s ; it is f a v o u r e d b y a h i g h s t r a i n rate (1,2). A c e r t a i n n u m b e r of p h e n o m e n o l o g i c a l c r i t e r i a have b e e n p r o p o s e d to d e t e r m i n e the c o n d i t i o n s of a p p e a r a n c e of this p h e n o m e n o n (i). In p a r t i c u l a r , Hart's c r i t e r i o n (3) a c c o u n t s for the e f f e c t of the local strain rate, w h i c h is n e g l e c t e d in ~ o n s i d e r e ' s c r i t e r i Q n (4). Recently, V i o l a n (5) p r o p o s e d a p h y s i c a l m e a n i n g to the p h e n o m e n o l o g i c a l p a r a m e t e r s of H a r t ' s criterion. A c c o u n t i n g for the fact that p l a s t i c d e f o r m a t i o n is t h e r m a l l y a c t i v a t e d and that the d e n s i t y of m o b i l e d i s l o c a t i o n s v a r i e s as d e f o r m a t i o n proceeds, we p r o p o u n d a more general f o r m u l a t i o n for the p r e c e d i n g criteria. Recall of H a r t ' s and V i o l a n c r i t e r i a In Hart's c r i t e r i o n (3), stress and strain are s u p p o s e d to y i e l d to the f o l l o w i n g r e l a t i o n ship : d~ = p dE + q d e (I) p and q b e i n g p a r a m e t e r s r e l a t e d to the state of the material. The s t a b i l i t y condition, w h i c h is e x p r e s s e d b y the fact that, as d e f o r m a t i o n proceeds, the d i f f e r e n c e o f the cross areas of the tensile s p e c i m e n does not tend to increase, can be w r i t t e n b y the f o l l o w i n g i n e q u a l i t y : Y+ m >I with

y= p/o

(2)

and m = Eq/o

Furthermore, V i o l a n (5) c o n s i d e r s the p l a s t i c d e f o r m a t i o n p r o c e e d s b y c r e a t i o n and m o t i o n of d i s l o c a t i o n s ; a h e t e r o g e n e o u s d e f o r m a t i o n w i l l be due to the e x i s t e n c e of a g r a d i e n t of d i s l o c a t i o n d e n s i t y a l o n g the specimen. The d e f o r m a t i o n is stable if this grad i e n t tends to ~ecrease. A t large deformation, this c o n d i t i o n is e x p r e s s e d as follows : 1 o

~ ~in~ 8e + ~ >1

(3)

V : a v e r a g e v e l o c i t y of the dislocations. T h i s r e l a t i o n s h i p can be i d e n t i f i e d w i t h equ. 2 : this allows g i v i n g a p h y s i c a l m e a n i n g to p a r a m e t e r s Y and m , w h i c h then r e p r e s e n t r e s p e c t i v e l y the s e n s i t i v i t y to s t r a i n (y=~o/o~e) and the s e n s i t i v i t y to s t r a i n rate (m = ~in~/~ine), a d m i t t i n g the d e n s i t y o f mobile d i s l o c a t i o n s is k e p t constant.

OO36-97~/78/121OS3-O~$02.OO/0 C o p y r i g h t (c) 197~ Pergamon P r e s s

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S t a b i l i t ~ c r i t e r i o n if d e f o r m a t i o n is t h e r m a l l ~ a c t i v a t e d In the h y p o t h e s i s of a t h e r m a l l y a c t i v a t e d jumping of o b s t a c l e s b y dislocations, the strain rate is y i e l d e d b y the relatioDship. = Pm K exp with

:

(- AG/kT)

(4)

K : constant A G : free G i b b s e n e r g y of a c t i v a t i o n of the jumping process. r e c t a n g u l a r barrier, it has the form :

In the case of a

AG = AG ~ ~v (5) o w h e r e a is the a p p l i e d stress and v the a c t i v a t i o n v o l u m e P m : d e n s i t y of m o b i l e dislocations, e x p r e s s e d b y G i l m a n ' s r e l a t i o n s h i p Pm=

(Pot + C E) exp

(- 6 e)

:

(6)

where P t is the i n i t i a l d i s l o c a t i o n s density, c o n s t a n t ; C is the mulo t i p l i c a t l o n c o e f f i c i e n t of dislocations, s u p p o s e d in a first a p p r o x i m a t i o n to v a r y as the inverse of g r a i n size (7) and 6 is the G i l m a n ' s a t t r i t i o n c o e f f i c i e n t of the m o b i l e f r a c t i o n of dislocations. (C and 6 are s u p p o s e d not to d e p e n d o n the a p p l i e d stress) U s i n g equ. 4, 5 and 6, w e can w r i t e d~

= k_TT d~ v~

- k_~T v

(

C Pot + Ce

- 6)

:

de

(7)

I d e n t i f y i n g this r e l a t i o n s h i p w i t h equ.l allows to e x p r e s s p a r a m e t e r s of H a r t ' s c r i t e r i o n ; this can b e f i n a l l y b e w r i t t e n as : k_TT (i - -- C + 6 )> i w P + Ce) ot or, for a g i v e n strain, u s i n g equ.4 kT

(i -

C Pot + Ce

p and q

(8)

: + 6) >

AG o

KP m - kT in( % ) e

(9)

T h e s e e x p r e s s i o n s show the influence of d i f f e r e n t e x p e r i m e n t a l p a r a m e t e r s on the a p p e a r a n c e of the i n s t a b i l i t y ; in p a r t i c u l a r , the test t e m p e r a t u r e and the a c t i v a t i o n volume (equ.8) or the s t r a i n rate (equ.9). C o m p a r i s o n w i t h the e x p e r i m e n t a l r e s u l t s E f f e c t of the t e m p e r a t u r e : As shown b y equ.8, a d e c r e a s e of the test t e m p e r a t u r e w i l l tend to m a k e the d e f o r m a t i o n m o r e q u i c k l y instable. T h i s is e f f e c t i v e l y o b s e r v e d in a v e r y low c a r b o n steel (ARMCO iron) b y V i o l a n (2), in the range of common s t r a i n rates. A s for us, we o b s e r v e d the same b e h a v i o u r (fig.l) in a H.S.L.A. steel at h i g h strain rates (~ = 400 s-l) (8) A t this rate the steel e x h i b i t s a d e f o r m a t i o n i n s t a b i l i t y b e l o w -50°C ; e q u i v a l e n t results h a v e b e e n b b s e r v e d , a t the same rate, in a m i l d steel (fig.2). E f f e c t of the s t r a i n rate : E q u . 9 shows the i n s t a b i l i t y appears earlier as the strain rate is higher. The c o n d i t i o n s of a p p e a r a n c e of the instability, as a f u n c t i o n of the t e m p e r a t u r e and s t r a i n rate have b e e n s t u d i e d in d e t a i l s b y V i o l a n (2) in A R M C O iron. F r o m r e l a t i o n 9, the p l a t of the i n s t a b i l i t y limit in a I/T vs in ~ d i a g r a m m u s t be a s t r a i g h t line, of slope -AG_/k . V i o l a n ' s r e s u l t s fit this a n a l y s i s v e r y w e l l for g r a i n sizes 25 and 17~m, and lead to a U v a l u e of AG O a r o u n d O.5eV, close to the v a l u e g i v e n b y C o n r a d (9) for t h e r m a l l y a c t i v a t e d j u m p i n g in iron or steel. E f f e c t of g r a i n size : D e f o r m a t i o n is g e n e r a l l y o b s e r v e d to be more stable as the grain is larger (iO) ; in o t h e r terms, the i n s t a b i l i t y range is s h i f t e d to lower t e m p e r a t u r e s or h i g h e r s t r a i n rates as the grains size i n c r e a s e s (2). R e s u m i n g the p r e c e d i n g l y m e n t i o n n e d a n a l y s i s of V i o l a n ' s results, w e can state that the slope of the s t r a i g h t lines, and therefore AGo, is larger as the g r a i n is finer.

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(7"(MPa) O- (MPa)

.

.

.

.

(~ 50Q

500

(~) - 50"C

(~

® -~oo'c ~)

_ 100" C _ 50"C

-150°C

(~)

T.A. t(ps) D

t (ps)

m 200

IOO

Fig.1

3 0

°

0

1 0' 0

2 'o 0

3 0' 0

Fig.2

E v o l u t i o n of stress vs time at d i f f e r e n t t e m p e r a t u r e s ; s t r a i n rate :e = 400 s -I. (Peak m e n t i o n e d b y the a r r o w n o t significant) : fig.1 : as r o l l e d H S L A steel, fig.2 : as r o l l e d m i l d steel. T h i s e f f e c t can be c a u s e d b y the e f f e c t of internal stress on A G : as a m a t t e r o of fact, the a p p l i e d stress ~ can be d i v i d e d in an i n t e r n a l stress o. and an e f f e c t i v e stress a c t i n g on the d i s l o c a t i o n s , o~. In the case of short r a n g e d o b s t a c l e ~ (low a c t i v a t i o n volume) o n l y o ~ c o n t r i b u t e s to the t h e r m a l l y a c t i v a t e d jumping. A G can t h e r e f o r e be w r i t t e n : o AG = AG' + u. v (I0) o o i AG' : h e i g h t of the p o t e n t i a l barrier. o The i n t e r n a l stress d e p e n d on the g r a i n size d, a c c o r d i n g to a d -I/2 law (12), in the same w a y as the scheme l e a d i n g to Hall and P e t c h ' s law. So, a d e c r e a s e of A G O is in fact o b s e r v e d if the g r a i n size is i n c r e a s e d ; as a c o n s e q u e n c e this d e c r e a s e of A G e n h a n c e s the Q d e f o r m a t i o n stability, as p r o v i d e d for in equ.9. In this equation, the g r a i n size also p l a y s a role in c o e f f i c i e n t C ; if d is increased, C decreases; c o n s e q u e n t l y the left h a n d e d term in equ.9 is increased, and, this way, the a p p e a r a n c e of the i n s t a b i l i t y delayed. To date, the r e s p e c t i v e p a r t s of the e f f e c t of the grain size on C and on AG can n o t be e v a l u a t e d q u a n t i tatively, o E f f e c t of the a c t i v a t i o n v o l u m e : A c c o r d i n g to equ.8, if the a c t i v a t i o n v o l u m e is lower, the d e f o r m a t i o n w i l l be m o r e stable. T h i s is a s c e r t a i n e d e x p e r i m e n t a l l y : at low t e m p e r a t u r e s and h i g h strain rates (fig.l and 2), the H S L A steel (very h i g h a c t i v a t i o n v o l u m e 28)) e x h i b i t s a m o r e i n s t a b l e b e h a v i o r t h a n the m i l d steel (low a c t i v a t i o n volume; a r o u n d 5Oh , b b e i n g the m o d u l u s of the B ~ r g e r s vector). In conclusion, the e q u a t i o n s 8 a n d 9 a c c o u n t w e l l for the c o n d i t i o n s of appearance of the d e f o r m a t i o n i n s t a b i l i t y in low c a r b o n steels and for the e f f e c t of thermal activ a t i o n on the p a r a m e t e r s of H a r t ' s criterion. It m u s t be s t r e s s e d this a n a l y s i s does not a c c o u n t for the e v e n t u a l e x i s t e n c e of p r e e x i s t i n g s u r f a c e defects, w h o s e e f f e c t s h a v e b e e n s t u d i e d b y J o n a s et al (12,13). For a given m e c h a n i c a l damage, it o n l y allows to p r e d i c t if the i n s t a b i l i t y limit is e a r l i e r of later, d e p e n d i n g on the i n f l u e n c e of the s t u d i e d m e t a l l u r g i c a l p a r a m e t e r s on c o e f f i c i e n t s y a n d m. B e c a u s e of the large n u m b e r of p a r a m e t e r s

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our a n a l y s i s r e m a i n s q u a l i t a t i v e a n d w o u l d therefore n e e d a more d e t a i l e d study. Acknowledgements

W e w i s h thank the L a b o r a t o r i e s of the R ~ g i e R e n a u l t for p r o v i d i n g the subject of this study and f u n d i n g it, as w e l l as P r o f e s s o r P.F. G o b i n for u s e f u l talks. References I - J.P. Poirier, (1976) 2 - P. Violan,

P l a s t i c i t ~ ~ h a u t e t e m p e r a t u r e des solides cristallins, Eyrolles, P a r i s

Scripta. Metall.,

7, 867

3 - E.W. Hart, Acta. Metall., 15, 351

(1973)

(1967)

4 - A. Considere, Ann. des P o n t s et Chauss~es, 9, n°6, 574, 5 - P. Violan, Scripta. Metall.,

6, 1175

(1972)

6 - J.J. Gilman, Appl. Mech. Rev., 21, 767, 7 - M.F. Ashby, Phil. Mag., 8 - G. Malaprade, 9-

21, 399

(1885)

(1968)

(1970)

D. Rouby, G. F a n t o z z i a n d P.F. G o b i n

(to be p u b l i s h e d in Mat. Sci. Eng,)

H. Conrad, in "The R e l a t i o n B e t w e e n the S t r u c t u r e s and M e c h a n i c a l P r o p e r t i e s of Metals", p. 475, Her M a j e s t y ' s S t a t i o n e r y Office, L o n d o n (1963)

10 - V. R a m a c h a n d r a n and E.P. A b r a h a m s o n ,

Scripta. Metall., 6, 287

11 - R.W. A r m s t r o n g in " D i s l o c a t i o n Dynamics", p Editors, Mc. Graw-Hill, N.Y., 1968 12 - J.J. JONAS,

(1972)

293, Rosenfield, Hahn, Bement, Jaffee,

R.A. H O L T and C.E. COLEMAN, Acta. Metall., 24, 911

13 - J.J. JONAS and B. BAUDELET, Acta. Metall., 25, 43

(1977)

(1976)

12