SCIENTIFIC AND TECHNICAL SECTION

A PHENOMENOLOGICAL TO HIGH-TEMPERATURE B. A. Movchan, O. G . K a s a t k i n ,

STRUCTURAL

APPROACH

CREEp L. M. Nerodenko, a n d E V. D a b i z h a

UDC 669.017:539.376

H i g h - t e m p e r a t u r e plastic deformation is d e s c r i b e d either via the mechanics of continuous media via f o r m a l models or by analysis of the e l e m e n t a r y p r o c e s s e s in an actual lattice. It is to be hoped that advances in r e s e a r c h on the m i c r o s c o p i c behavior of solids at high t e m p e r a t u r e s will provide an approach that will combine to s o m e extent m e c h a n i c a l and physical concepts, for instance r e p l a c e m e n t of a continuous medium by a body with d i s c r e t e s t r u c t u r a l e l e m e n t s . However, the analysis of the r e s p o n s e to load and t e m p e r a t u r e is to be based not on a p a r t i c u l a r e l e m e n t a r y p r o c e s s but on a s t a t i s tical approach, via the a v e r a g e p r o p e r t i e s of a l a r g e a s s e m b l y of identical elements, e.g., when e l e m e n t a r y s h e a r is considered [1] as motion of a flow unit involving o v e r c o m i n g an e n e r g y b a r r i e r , while s t e a d y - s t a t e c r e e p is considered f r o m the t h e o r y of r a t e p r o c e s s e s , as in [2]. Here we use a previous study [3] with a s t r u c t u r a l phenomenological model for a solid, which reflects the basic features of h i g h - t e m p e r a t u r e plastic deformation. A deformable body contains r a n d o m l y distributed s o u r c e s (centers) of density N 0. Each s o u r c e is independent and has a p a r t i c u l a r activation s t r e s s . The set of these s t r e s s e s f o r m s a c e r t a i n range with a lower bound a0 and a mean ~. A m a c r o s c o p i c s t r e s s a exceeding cr o activates the s o u r c e s after t i m e s t; the individual values t2, . . . , t n f o r m a m e a n t (the mean time to activation under s t r e s s ) . An activated s o u r c e produces m i c r o s c o p i c plastic deformation (6 is the mean s h e a r due to a single s o u r c e ) . Each s o u r c e then c e a s e s to operate on account of accumulating internal s t r e s s . If the t e m p e r a t u r e is high enough, t h e r m a l l y activated r e c o v e r y o c c u r s in a m e a n time T (mean r e c o v e r y time). A s o u r c e after r e c o v e r y can be r e a c t i v a t e d , and so on. These conditions do not impose any specific r e s t r i c t i o n s on the s o u r c e s ; they m e r e l y r e f l e c t the phenomenological sequence of m i c r o s c o p i c stages in h i g h - t e m p e r a t u r e plastic deformation f r o m the time when cr is applied: plastic deformation, local hardening, r e c o v e r y . We denote the numbers of c u r r e n t l y activated and r e c o v e r e d s o u r c e s by Ncl and Nrl r e s p e c t i v e l y to put

No-->-Nc, -+ Nr,. After r e c o v e r y , the Nrl s o u r c e s will c o r r e s p o n d to the initial N0ones , in p a r t i c u l a r as r e g a r d s activation s t r e s s . On reactivation they make an additional contribution to the plastic deformation. The cycle can probably be repeated s e v e r a l t i m e s , m o s t likely in parallel with p r o c e s s e s initiating failure (pore nucleation, growth, and coalescence). The total plastic deformation e at time t after application of a is e = ~.Nc,

(1)

where Nc is the sum of all c u r r e n t activations (initial and repeated). L

Fig. i. Characteristic curves.

creep

It has been shown [3] that, by analogy with dislocation multiplication, s o u r c e activation rate under a constant s t r e s s in the absence of r e c o v e r y can be r e p r e s e n t e d as

E. O. Patch Electric Welding Institute, Academyof Sciences of the Ukrainian SSR. ~ranslated from Problemy Proehaosti, No. 9, pp. 3-9, September, 1974. Originalarticle submitted November 11, 1973.

9 19 75 Plenum Publishing Corporation, 227 West 1 7th Street, New York, N. Y. 10011. No part o f this publication may be reproduced, stored in a retlieval O'stem, or transmitted, in any fi)nn or by an), means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15.00.

1041

I

-, o

I

I

f

"/z 2,C

Fig.2.

/

f

Y.

I

J

tel - Gq9 Fe

f Io 20 30 #0 t, min Typical creep curve for nickel.

$

o.4

0,8

p,2

t,s

2,0 t, h

F i g . 3. C r e e p c u r v e s f o r n i c k e l a l l o y s

dNnc

d~

--

(2)

rNnc,

w h i c h is a d i f f e r e n t i a l e q u a t i o n f o r a s i m p l e f i r s t - o r d e r r e a c t i o n . H e r e Nnc is the n u m b e r of u n a c t i v a t e d s o u r c e s , w h i l e r is t h e r a t e c o n s t a n t , w h i c h is g e n e r a l l y d e p e n d e n t on t e m p e r a t u r e , s t r e s s , and s t r u c t u r e , w i t h 1 / r = t the m e a n l i f e t i m e of an a c t i v a t e d s o u r c e a f t e r the s t r e s s h a s b e e n a p p l i e d . It is l o g i c a l to s u p p o s e t h a t r e c o v e r y of a 6rice a c t i v a t e d s o u r c e a l s o o b e y s a s i m p l e f i r s t - o r d e r l a w , and s o in the g e n e r a l c a s e a s t r e s s a p r o d u c e s a c t i v a t i o n , r e c o v e r y , a c t i v a t i o n , e t c . , and s o we c a n w r i t e t h a t dNnc

= - - rNnc + s ( N - - Nnc),

dt

(3)

w h e r e N is the i n i t i a l n u m b e r of s o u r c e s t h a t w i l l be a c t i v a t e d b y ~, s is t h e r e c o v e r y r a t e c o n s t a n t , w h i c h is d e p e n d e n t on t e m p e r a t u r e , s t r e s s , a n d s t r u c t u r e , w i t h 1 / s = T. We i n t e g r a t e (3) w i t h Nnc = N f o r t = 0, which gives Nnr -- - N - -? .L ~ _-$- s (1 - - e-(r+')t ) + N e -('+`~t. T h e t o t a l n u m b e r of a c t i v a t i o n s is t

No =

r.N, oet.

(4)

0

We i n t e g r a t e (4) and s u b s t i t u t e f o r N c in (1) to g e t t h e e x p r e s s i o n f o r the p l a s t i c d e f o r m a t i o n z = -6N [

rs

i -t- ~ rS

(i - - e-(r+s)t)].

(5)

It has b e e n shown t h a t 6 N 0 ( 1 - e - ( a - a 0 ) / ~ ) c o r r e s p o n d s to the l i m i t i n g p l a s t i c d e f o r m a t i o n ( t ~ ) arising in r e s p o n s e to a m a c r o s c o p i c s t r e s s a w h e n t h e r e is no r e c o v e r y (s = 0). T h e r e f o r e , we f i n a l l y have

8 = 8 1 ~ rs

t+~-(l

__ e_(r+~)t)].

r~

(6)

T h e n h i g h - t e m p e r a t u r e p l a s t i c d e f o r m a t i o n at c o n s t a n t a is d e t e r m i n e d b y t h r e e p a r a m e t e r s : ~1, the d e f o r m a t i o n c a p a c i t y of the m a t e r i a l in t h e a b s e n c e of r e c o v e r y , and t h e k i n e t i c c o e f f i c i e n t s r and s, w h i c h c h a r a c t e r i z e r e s p e c t i v e l y the r a t e s of the p r o c e s s e s of h a r d e n i n g and r e c o v e r y . In g e n e r a l , the t i m e d e p e n d e n c e of the d e f o r m a t i o n t a k e s the f o r m of a c o n v e x c u r v e w i t h a s t r a i g h t - l i n e s e c t i o n f o r t l a r g e enough, w h e n the s e c o n d t e r m in (6) b e c o m e s i n f i n i t e l y s m a l l . E x a m p l e s of s u c h c u r v e s a r e g i v e n in F i g . 1. W h e n the t e m p e r a t u r e is low enough, we have s ~ 0, a n d (6) t r a n s f o r m s for a decaying plastic deformation for a = coast:

into a s t a n d a r d r e l a t i o n s h i p

s = el (i - - e-rt). We find the g e n e r a l r e l a t i o n s h i p f o r the c r e e p r a t e b y d i f f e r e n t i a t i n g (6):

~

+ ~

e-C~+~)tl.

(7)

F r o m (7) w e g e t t h e f o l l o w i n g p a r t i c u l a r c a s e s : a is the i n i t i a l c r e e p r a t e (t = 0):

~n = st'r = 8---1i '

1042

(8)

\

f,h

Ni-54 Vo/h

I

~', %

L -

~25,2

-

kgf/mmy

2,t o,6'

~4

I

o'~ Ni

4r -4~ " ~ " "4

~"~'~----"~ 20

40

60

80

ire

0

I00

F i g . 4. Values of e l , ey, t, and ~ for Ni-Fe alloys 9

F i g . 5.

22,!

200

300

400

t, h

C r e e p c u r v e s f o r Khl7N1LM3 heat-resisting steel.

while b is the s t e a d y - s t a t e c r e e p r a t e when t is s u f f i c i e n t l y l a r g e and the s e c o n d t e r m in (7) tends to z e r o : 9

F.S

gl

Sy=el r +------T-- /-+ ~

(9)

If t- << u we have a p p r o x i m a t e l y that ~y =

_-=-,

(10)

T

i.e., the r e c o v e r y will c o n t r o l the r a t e of s t e a d y - s t a t e c r e e p .

If ~ << t-, then

s

ey = _z,

(Ii)

t

i.e., the c r e e p r a t e will be d e t e r m i n e d by f e a t u r e s of the p l a s t i c d e f o r m a t i o n . T h e r a t i o of t h e s e c r e e p r a t e s is found f r o m (8) and (9)." s--~-i= I - 4 - -r7 = 1~ ~y

_ . t

(12)

T h i s r a t i o r e f l e c t s the f o r m of the c r e e p c u r v e and g i v e s i n f o r m a t i o n about the r e l a t i v e values of u and t-. F i g u r e 1 shows s o m e c h a r a c t e r i s t i c f o r m s of c r e e p c u r v e s r e f l e c t i n g r / t ; c u r v e 1 c o r r e s p o n d s to c o n d i tions w h e r e r > t and ~i_>" gy_. S i m i l a r r e l a t i o n s h i p s apply f o r p u r e m e t a l s at t e m p e r a t u r e s of 0 . 5 - 0 . 8 Trap. Curve 2 c o r r e s p o n d s to r ~ t and g i ~ 2gy, while c u r v e 3 c o r r e s p o n d s to r < t and ~i ~ iv- The l a t t e r r e l a t i o n s h i p s a r e c h a r a c t e r i s t i c of c r e e p in p u r e m e t a l s at high t e m p e r a t u r e s [4]. We p u t ~ i / ~ y - 1 = a o r = ~7

(13)

and s u b s t i t u t e (13) and (9) into (6) to get f o r the s t e a d y - s t a t e s t a g e that -

--

8 --

~'yl

.

(a A- 1)2,

(14)

s 1

w h e r e e is the d e f o r m a t i o n c o r r e s p o n d i n g to t i m e t in the s t e a d y - s t a t e c r e e p . T h e n we determine_ _ ~i and ~y f r o m the c r e e p c u r v e and use (13), (14), and (9) to c a l c u l a t e the above p a r a m e t e r s e l , T, t. This d e s c r i p t i o n of h i g h - t e m p e r a t u r e c r e e p does not take into a c c o u n t the i n s t a n t a n e o u s d e f o r m a t i o n a r i s i n g at the instant of loading, so in quantitative c a l c u l a t i o n s on e l , T a n d t we have to c o n s t r u c t the c u r v e as c o r r e c t e d f o r the i n s t a n t a n e o u s d e f o r m a t i o n . F o r c e r t a i n t y p e s of c u r v e , f o r w h i c h T > t ( c u r v e 1 in F i g . 1), one can get s o m e u n c e r t a i n t y in d e f i n ing the r e g i o n of instantaneous d e f o r m a t i o n and in c h o o s i n g the c o o r d i n a t e o r i g i n , and a l s o the o r i g i n f o r d e t e r m i n i n g e i ; in t h e s e c a s e s one needs to u s e t r i a l and e r r o r , s p e c i f y i n g that e l , t, r d e s c r i b e as b e s t as p o s s i b l e the o b s e r v e d c u r v e in a c c o r d a n c e with (6). F i g u r e 2 shows the c r e e p c u r v e f o r p u r e nickel at 800~ and 4 k g / m m 2 ; the p u r e nickel was p r o d u c e d by a twofold e l e c t r o n - b e a m m e l t i n g of c a t h o d e s of g r a d e NO. The c a s t i n g s of d i a m e t e r 100 tuna w e r e f o r g e d and r o l l e d with i n t e r m e d i a t e a n n e a l i n g at 800~C. A f t e r the final d e f o r m a t i o n , w h i c h w a s 50%, we p r e p a r e d

1043

T=600';C(5=/,3'kgf/m

kgf/mm~

".z.l

i

2

-2

-/

2

0

3 bTt~min)

r=8oo*cf]=l,2'kgf/r m2)

~'~.~,, ' ""~

~

soo~

oobro,5

r

4 5 In (f, min) b F i g . 6. R e l a t i o n s h i p s _ f o r p u r e n i c k e l : a) t - c r ; b) r - - a . o

/

2

3

f i a t s p e c i m e n s of t h i c k n e s s 1 m m and of l e n g t h 10 m m in the working part. Before creep testing, the specimens were subj e c t e d to r e c r y s t a l l i z a t i o n a n n e a l i n g u n d e r v a c u u m at 900~C f o r 1 h. T h e c u r v e w a s r e c o r d e d w i t h an e q u i p m e n t f o r h i g h - t e m p e r a t u r e p r e c i s i o n t e s t i n g f o r c r e e p and l o n g - t e r m s t r e n g t h u n d e r v a c u u m [5]. T h e e q u i p m e n t w a s i n t e n d e d f o r p e r f o r m i n g t e s t s w h i l e m a i n t a i n i n g a c o n s t a n t s t r e s s on t h e s p e c i m e n , and it e n a b l e s one to m e a s u r e the s t r a i n w i t h high s e n s i t i v i t y , w h i l e a u t o m a t i c a l l y r e c o r d i n g the c r e e p c u r v e . T h e i n s t a n t a n e o u s c r e e p d e f o r m a t i o n w a s 2% and w a s t a k e n into a c c o u n t in c o n s t r u c t i n g the g r a p h . C a l c u l a t e d v a l u e s a r e s h o w n a s p o i n t s on the s o l i d l i n e ( e x p e r i m e n t a l c u r v e ) , , t h e s e b e i n g d e r i v e d b y the a b o v e m e t h o d : the v a l u e s o f ~ i and ey w e r e d e t e r m i n e d f r o m the e x p e r i m e n t a l c u r v e , w h i l e the c a l c u l a t i o n s of t, T, e t w e r e p e r f o r m e d f r o m (9), (13), and (14) o r b y a c o m p u t e r . S a t i s f a c t o r y a g r e e m e n t b e t w e e n t h e o b s e r v e d and c a l c u l a t e d v a l u e s w a s o b t a i n e d w i t h the f o l l o w i n g v a l u e s f o r t h e p a r a m e t e r s : a t = 3.19%; T = 58.8 r a i n and t = 6.5 r a i n . To c o n f i r m t h i s a p p r o a c h and m e t h o d of c a l c u l a t i o n w e p r o c e s s e d the p r i m a r y p e a k c u r v e s f o r v a r i o u s m e t a l s and a l l o y s a s g i v e n in t h e l i t e r a t u r e . F i g u r e 3 s h o w s c r e e p c u r v e s f o r n i c k e l - i r o n a l l o y s [6] f o r 900~C and 4 k g / m m Z ; the c a l c u l a t e d p o i n t s a r e shown, w h i l e the s o l i d l i n e s a r e f r o m e x p e r i m e n t .

F i g u r e 4 s h o w s the c a l c u l a t e d v a l u e s of e 1, T, t f o r N i - F e a l l o y s ; t h e s e p a r a m e t e r s w e r e o b t a i n e d f o r p u r e n i c k e l by c a l c u l a t i o n f r o m o u r own e x p e r i m e n t a l r e s u l t s . F o r c o m p a r i s o n , t h e f i g u r e s h o w s t h e relatioashi_p of t h e s t e a d y - s t a t e c r e e p r a t e ~y to the i r o n c o n c e D t r a t i o n . It is c l e a r t h a t the m a x i m u m i n c r e a s e in T o c c u r s when the i r o n c o n c e n t r a t i o n i n c r e a s e s f r o m 0 to 30%, w i t h h a r d l y a n y c h a n g e in T f o r h i g h e r i r o n c o n t e n t s . The m a x i m u m on the c u r v e r e l a t i n g { to F e c o n c e n t r a t i o n a l s o c o r r e s p o n d s to r o u g h l y 20-40%. T h e s e a l l o y s a l s o have the h i g h e s t c r e e p r e s i s t a n c e (rain ~y), a n d a r e found [7] to h a v e the h i g h e s t a c t i v a t i o n e n e r g y f o r c r e e p . T h e m a x i m u m v a l u e s of u and t - c o r r e s p o n d in t h e F e - N i c o n c e n t r a t i o n d i a g r a m to the F e N i 3 s u p e r l a t t i c e r e g i o n . F i g u r e 5 s h o w s a d e s c r i p t i o n o f the c r e e p c u r v e f o r h e a t - r e s i s t i n g s t e e l c o n t a i n i n g 17% C r ; 11% Ni; 2.5% Mo at 600~C and s t r e s s e s of 22.1 and 25.2 k g / m m 2 [8]. T h e c a l c u l a t e d v a l u e s (points) a g r e e S a t i s f a c t o r i l y w i t h t h e e x p e r i m e n t a l v a l u e s w i t h t h e f o l l o w i n g v a l u e s f o r the p a r a m e t e r s : f o r a = 25.2 k g / m m 2, e 1 = 0.71%, t- = 9.26 h, ~ = 167.5 h; f o r a = 22.1 k g / m m 2, e 1 = 0.35?o, t- = 17.66 h, ~" = 164.2 h. T h e p a r a m e t e r s r = l / t - and s = 1 / r a r e d e p e n d e n t on t e m p e r a t u r e , s t r e s s , and s t r u c t u r a l s t a t e of the m a t e r i a l ; a n a l o g y w i t h the r a t e c o n s t a n t s of r e a c t i o n s i n d i c a t e t h a t r ~ e - (U1/flT) and s ~ e - ( U 2 / R T ) , w h e r e U 1 and U 2 a r e the a c t i v a t i o n e n e r g i e s f o r h a r d e n i n g and s o f t e n i n g r e s p e c t i v e l y . In g e n e r a l , U i ~ U 2. T h e n the e x p r e s s i o n s f o r t - a n d u m a y be put a s U,

't = toe R"-~f (o'); U=

:~

~~e RT f (a),

w h e r e f(~) is s o m e f u n c t i o n of s t r e s s , w h i l e t o and T Oa r e c o n s t a n t c o e f f i c i e n t s of a g i v e n s t r u c t u r a l s t a t e of the m a t e r i a l . A c t i v a t i o n and r e c o v e r y in s o u r c e s o c c u r d u r i n g t h e a c t i o n of a, s o t - a n d u s h o u l d be p r o p o r t i o n a l to t h e p r o b a b i l i t y t h a t the s o u r c e s do not b e c o m e a c t i v a t e d , P0. It has b e e n shown [3] t h a t P0 = e

~

,

w h e r e a0 and ~ a r e c o r r e s p o n d i n g l y the m i n i m a l a n d m e a n s t r e s s e s We a s s u m e t h a t f(a) = e - ( a - a 0 ) / ~

to g e t Us

-t=to eRre

1044

f o r a c t i v a t i o n of a s o u r c e .

~--Oe

~ ;

(15)

Ut

~-.oo

7 -----gee Rr e

a

(16)

Figure 6a shows lnt = f(e) for pure nickel at various temperatures; the observations are well fitted by sections of straight lines in semilog plots, which confirms that we have an exponential relationship of tto stress. The numerical values of ~ are given in parentheses; construction of.the in~ = f (~) curve of Fig. 6b gives a similar relationship. An additional confirmation of (15) and (16) comes by substituting from (9), (10), and (11); we substitute (15) and (16) into (9) with Ui = U2 = U to give ~--Oo

9

ey---- ~tNo(l--e

5

U (~--ffi

o'--o=

(to+%)e

~Noe ~r (e ~ --1).

) U

(17)

to _~- iO

~ err

We can neglect the unity for l a r g e and m e d i u m values of ~, i.e., U =

EOe

o

(18)

RT e ~ ,

where 8o = to-~----7 e

"

~(to-~o)e"

F o r a s m a l l (1 < ( a - a 0 ) / a -< 3-4), the f a c t o r e ( a - a 0 ) / a - 1 in (17) can be a p p r o x i m a t e d a c c u r a t e l y by the function Aan/F~ n, where A is a c e r t a i n constant coefficient. Then (18) can be put as follows: U Ey =

~.,le

(19)

RT on '

where el - -

A6N~

(to + %) (j)n

F o r p a r t i c u l a r c a s e s of (10) and (11) we get r e l a t i o n s h i p s s i m i l a r to (18) and (19), the only difference being that the values of ~0 and ~1 will contain instead of the factor (t o + To) the s i m p l e coefficients T o or t~ respectively. Equations (18) and (19) a r e the s a m e as the f a m i l i a r e m p i r i c a l r e l a t i o n s h i p s c o m m o n l y used in des c r i b i n g the r a t e of s t e a d y - s t a t e c r e e p at high, m e d i u m , and low levels of applied s t r e s s . This m a y be c o n s i d e r e d a c o n f i r m a t i o n of the a s s u m p t i o n s m a d e in (15) and (16). In the above s i m p l e function of vation e n e r g i e s by p e r a t u r e variation

c a s e s , ~ 0 is hardly dependent on t e m p e r a t u r e , while ~1 is only weakly so, s i n c e ~r is a Young's modulus [9]. T h e r e f o r e , (18) and (19) can be used in calculating the t r u e a c t i balancing out the f a c t o r e (cr/~) in (18) [9, 10] and introducing a c o r r e c t i o n for the t e m of Young's modulus in (19).

If U1 # U2, (18) and (19) contain not the s i m p l e s u m (t o + To) but the f a c t o r (t o + T0e(U2-UI)/RT), i.e., a0 and ~1 will be f a i r l y s t r o n g l y dependent on t e m p e r a t u r e . T h e r e f o r e , application of (18) and (19) to p r o c e s s i n g of e x p e r i m e n t a l data will give s o m e fictitious activation e n e r g y that does not r e f l e c t the t r u e t h e r m a l l y a c t i v a t e d p r o c e s s e s of p l a s t i c d e f o r m a t i o n . In that c a s e , U1 and U2 have to be d e t e r m i n e d f r o m the t e m p e r a t u r e dependence of t-and T. It is notable that (9), (18), and (19) can be derived also f r o m the following s t a n d a r d relation: 0e 9

at

8 =

~

v

-

-T,

(20)

Os

w h e r e v is the r e c o v e r y r a t e for the internal s t r e s s and h is the d e f o r m a t i o n hardening f a c t o r . Since oo a--f = ~

aa

aN "-~-'

a~

aa

a---Y- =

a'-W- " a---~"

aN

1045

aN

Ti '

:

"

then

8:

8

aN .-~- .

The r e c o v e r y r a t e vr of activated s o u r c e s can be r e p r e s e n t e d as follows: vr = yaN

= s (N

-

-

Nne).

We use the above e x p r e s s i o n for Nnc and get the r e c o v e r y r a t e in the s t e a d y stage (t--**o) as vr = N

sr $-~r

"

Then (20) can be t r a n s f o r m e d to =

5N

sr

. =

s+r

e~

i+:c

'

which is identical with (9), (18), and (19). Equations (15) and (16) f o r ' t and T enable us to draw s o m e conclusions on e i / e y . We substitute (15) and (16) into (12) with U1 = U2 to get en =

(21)

cey,

where TO

c=t+To

=c~

Then (21) is m e t when one is c o m p a r i n g experin.~.ental data. F o r instance, it has been shown [4] that for the c r e e p of stainless steel at 704-829*(2 we have e i = 3.3 ~y, i.e., c = 3.3. CONCLUSIONS 1. Relationships have been derived within a s t r u c t u r a l phenomenological a p p r o a c h to d e s c r i b e hight e m p e r a t u r e p l a s t i c deformation; e x a m p l e s have been taken for p u r e nickel, n i c k e l - i r o n alloys, and c h r o m i u m - nickel s t e e l to confirm the model and the method of calculation. 2.

H i g h - t e m p e r a t u r e plastic deformation at constant s t r e s s is d e t e r m i n e d by the following: a) the kinetic coefficients {-and u which c h a r a c t e r i z e the r a t e s of hardening and r e c o v e r y together with the exponential dependence on s t r e s s and t e m p e r a t u r e ; b) the deformation capacity of the m a t e r i a l e 1 in the absence of r e c o v e r y .

3. Methods have been p r o p o s e d for calculating the t r u e activation e n e r g i e s of hardening and r e c o v e r y , as well as the m i c r o s c o p i c elastic limit a. LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

1046

CITED

W. Kauzmann, T r a n s . AIME, 143 (1941). A . S . K r a u s z , J . Appl. Phys., 12, No.6 (1971). B . A . Movchan and L. M. Nerodenko, P r o b l e m y P r o c h n o s t i , No. 10 (1973). F . Garofalo, C r e e p and L o n g - T e r m Strength Laws for Metals and Alloys [Russian translation], Metallurgizdat, Moscow (1968). E . V . Dabizha, P r o b l e m y P r o c h n o s t i , No. 3 (1971). S. K a r a s h i m a , T. Motomiya, and H. Oikawa, Technology R e p o r t s , Tokyo U n i v e r s i t y , 33, No.2 (1968). K . A . Osipov and Tian T e - c h ' e n g , izv. Akad. Nauk SSSR, OTN, M e t a l l u r g i y a i Toplivo~-No.2 (1961). J . A . M a z z a and G. Willoughby, J. Iron Steel Inst., 204, No. 7 (1966). B . A . Movchan, Ftz. Metal. Metal., 24, No. 6 (1967). B . A . Movchan and L. M. Nerodenko, F i z . - K h i m . Mekh. M a t e m . , No. 6 (1967).