ANALYSIS

OF

SPASMODIC

G.

THE

MECHANISM

OF

DEFORMATION

A.

Malygin

UDC 539.5.015.1

Ever since Cottrell suggested that the unstable, s p a s m o d i c deformation of interstitial and substitutional alloys (the P o r t e v i n - - L e Chatelier effect [1]) was a s s o c i a t e d with the blocking of moving dislocations by dynamic impurity a t m o s p h e r e s (dynamic s t r a i n aging) [2, 3], e x p e r i m e n t a l w o r k has continued in this field [4-21] and a g r e a t deal of experimental m a t e r i a l has been accumulated, so facilitating a m o r e differentiated and detailed analysis of this phenomenon than had e a r l i e r been feasible. If we gradually r a i s e the t e m p e r a t u r e of the experiment, keeping the d e f o r m a t i o n rate constant, then on plotting the s t r a i n d i a g r a m s of dilute interstitial and concentrated substitutional solutions we shall obtain s t r a i n curves in the sequence indicated in Fig. 1. For v e r y low t e m p e r a t u r e s the deformation is stable (curve 1). On reaching a certain t e m p e r a t u r e it becomes unstable, but reraains homogeneous (curve 2), or quasihomogeneous if many bands of nonuniform deformation a r i s e at the s a m e time. At still higher t e m p e r a t u r e s the s a m p l e s t a r t s deforming inhomogeneously: a yield tooth and plateau a r e formed, with the Luders front propagating along the s a m p l e (curve 3). On f u r t h e r raising the t e m p e r a t u r e the d e f o r m a t i o n on the plateau b e c o m e s unstable: b r e a k s in the load appear (curve 4); the number of discontinuities i n c r e a s e s and the length of the plateau cont r a c t s , until finally the difference between the plateau and the next strengthening curve is h a r d to distinguish (curve 5). The Luders front then exhibits a r r e s t s in its motion. The a p p e a r a n c e of each such jump involves the development of a new C h e r n o v - - L u d e r s band (curve 4). At a still higher t e m p e r a t u r e , c o r r e s p o n d i n g to the region of dynamic s t r a i n aging, load diseontinuities (curve 6) set in as a r e s u l t of the periodic p a s s a g e of P o r t e v i n - - l e Chatelier bands along the sample, these being analogous to the Luders bands but having a different m e c h a n i s m of f o r m a t i o n . * In the lowt e m p e r a t u r e range of the region of dynamic aging the bands propagate smoothly~ The vanishing of a band at one edge of the sample is accompanied by the development of a new band at the opposite edge and a c o r responding reduction in load [8-11, 15, 16] (Fig. 2)~ F u r t h e r raising the t e m p e r a t u r e causes slight oscillations in the load between the c h a r a c t e r i s t i c discontinuities of the A type. With increasing t e m p e r a t u r e the amplitude of these oscillations i n c r e a s e s . Type B (carve 7 in Fig. 1) is distinguished by the fact that the r i s e and fall times of the load a r e approximately equal (Fig. 2). With rising t e m p e r a t u r e , the spasmodic d e f o r m a t i o n of type B p a s s e s into type C (curve 8 in Fig. 1), a special c h a r a c t e r i s t i c of which is a rapid fall in load and a r e l a t i v e l y slow r e s t o r a t i o n (Fig. 2). On the section c o r r e s p o n d i n g to the r i s e in s t r e s s the s a m p l e d e f o r m s elastically (or apprordmately so). Each discontinuity in the load is a s s o c i a t e d with the generation of a new band, which, not being able to p r o p a gate, stops; alongside it in the undeformed region yet another band is c r e a t e d . This p r o c e s s continues until the bands c o v e r the whole sample, after which they s t a r t moving f r o m the edge of the s a m p l e at which they were originally g e n e r a t e d . Thus for s p a s m o d i c d e f o r m a t i o n of type C the bands move in an *In certain cases Luders and P o r t e v i n - - L e Chatelier bands develop sinmltaneously in the sample, as a r e s u l t of which a c h a r a c t e r i s t i c "ripple" appears on the r i s i n g p a r t s of the teeth on the yield plateau (curve 4 in Fig. 1). A. F. Ioffe Physicotechnical Institute, A c a d e m y of Sciences of the USSR, Leningrad. T r a n s l a t e d from P r o b l e m y Prochnosti, No. 2, pp. 12-18, F e b r u a r y , 1975. Original a r t i c l e submitted May 22, 1973.

9 1975 Plenum Publishing Corporation, 227 West 17th Street, New l~brk, N. Y. 100l l. No part o f this publication may be reproduced, stored in a retrieval system, or tralzsmitted, iJt any fi~rm or by an3' means, electronic, mechanical, photocopying, microfilmh~g, recording or otherwise, without written permission o f the publisher. A copy o f this article is" available from the publisher [or $15.00.

136

F i g . 1. V a r i o u s c a s e s of u n s t a b l e d e f o r m a t i o n .

u n s t a b l e m a n n e r , i n c o n t r a s t to t h e m o t i o n of t h e b a n d s f o r t y p e A . S p a s m o d i c d e f o r m a t i o n of t y p e B i s i n t e r m e d i a t e b e t w e e n A a n d C: a f a l l i n l o a d only c a u s e s t h e b a n d to m o v e m o r e s l o w l y , without a c t u a l l y stopping. Further raising the temperature ' stably and homogeneously.

c a u s e s t h e j u m p s to v a n i s h ( c u r v e 9 in F i g . 1); t h e s a m p l e d e f o r m s

It s h o u l d be n o t e d t h a t t h e u n s t a b l e m o t i o n of t h e P o r t e v i n - - L e C h a t e l i e r b a n d s only o c c u r s f o r a c o n s t a n t d e f o r m a t i o n r a t e (a " h a r d " m a c h i n e ) . In e x p e r i m e n t s with a c o n s t a n t l o a d i n g r a t e ("soft" m a c h i n e ) t h e b a n d s m o v e s m o o t h l y [18, 19]. W e m a y t h u s c o n c l u d e t h a t s p a s m o d i c d e f o r m a t i o n s of t y p e s B and C a r e a s s o c i a t e d with t h e l o a d i n g c o n d i t i o n s . E s s e n t i a l l y t h e r e g i o n of u n s t a b l e and i n h o m o g e n e o u s d e f o r m a t i o n , i n c l u d i n g t h e c a s e s d e s c r i b e d by c u r v e s 6 - 8 , i . e . , t h e r e g i o n of d y n a m i c s t r a i n a g i n g , c o n s t i t u t e s t h e P o r t e v i n - - L e C h a t e l i e r e f f e c t . T h e o t h e r c a s e s i l l u s t r a t e d ( c u r v e s 2 - 4 in F i g . 1) r e l a t e to v a r i o u s m o d i f i c a t i o n s of t h e p h e n o m e n o n , i n c o r p o r a t i n g a s h a r p l i m i t and a y i e l d p l a t e a u , a n d in t h e s o l u t i o n s u n d e r c o n s i d e r a t i o n t h e y a r e a s s o c i a t e d with t h e f o r m a t i o n of s t a t i c a t m o s p h e r e s of i m p u r i t y a t o m s . In d i l u t e s u b s t i t u t i o n a l s o l u t i o n s in f a c e c e n t e r e d c u b i c m e t a l s t h e y i e l d t o o t h and p l a t e a u a r e a b s e n t , and only t h e c l a s s i c a l P o r t e v i n - - L e C h a t e l i e r effect is observed. In t h i s p a p e r we s h a l l p r e s e n t a t h e o r e t i c a l a n a l y s i s of the P o r t e v i n - - L e C h a t e l i e r c o n d i t i o n s with d u e a l l o w a n c e f o r t h e v e l o c i t y d e p e n d e n c e of t h e y i e l d s t r e s s , t h e i n h o m o g e n e o u s c h a r a c t e r of t h e s t r a i n d i s t r i b u t i o n in t h e s a m p l e , and t h e d y n a m i c p r o p e r t i e s of t h e t e s t m a c h i n e . D y n a m i c R e t a r d i n g F o r c e of t h e A t m o s p h e r e . W h e n d y n a m i c a t m o s p h e r e s of d i s s o l v e d a t o m s a r e f o r m e d a r o u n d m o v i n g d i s l o c a t i o n s , t h e t o t a l r e s i s t a n c e e x p e r i e n c e d by a d i s l o c a t i o n in i t s m o t i o n is "r = "r" + 1:~+ "ra - - ~t,,"

(1)

H e r e r i = r i ( e ) i s t h e i n t e r n a l l o n g - r a n g e s t r e s s ; r * i s t h e c o m p o n e n t of y i e l d s t r e s s d e p e n d i n g on the t e m p e r a t u r e and d e f o r m a t i o n v e l o c i t y in a c c o r d a n c e ~vith t h e e x p r e s s i o n f o r a 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 =

~0 e

kr

,

(2)

w h e r e H i s t h e a c t i v a t i o n e n t h a l p y . T h e s t r e s s Tin = (M/K')T is a s s o c i a t e d ~ [ t h t h e a c c e l e r a t i o n of t h e m o v a b l e p a r t s of t h e m a s s M of t h e m a c h i n e , w h i c h h a s a r i g i d i t y K ' . T h e d y n a m i c f r i c t i o n a l f o r c e ~'d a r i s i n g f r o m t h e p r e s e n c e of t h e a t m o s p h e r e m a y be e x p r e s s e d in t h e f o r m [3, 14]

Td(vd, t) = ~S (C--C~

dx,S~, b~ '

(3)

w h e r e u = (A/r)sin~0 i s t h e i n t e r a c t i o n e n e r g y b e t w e e n t h e i m p u r i t y a t m o s p h e r e and a n e d g e d i s l o c a t i o n r e s u l t i n g f r o m s t e r i c m i s m a t c h (A i s t h e i n t e r a c t i o n c o n s t a n t ) ; 2 is t h e a t o m i c v o l u m e of t h e s o l v e n t , b

F i g . 2. V a r i o u s t y p e s of s p a s m o d i c d e f o r m a t i o n (A, B, C).

137

i s t h e B u r g e r s v e c t o r , vd is t h e v e l o c i t y of t h e d i s l o c a t i o n , t i s t h e t i m e , c o is t h e m e a n a t o m i c c o n c e n t r a t i o n of t h e i m p u r i t y a t o m s . T h e d i s t r i b u t i o n and c o n c e n t r a t i o n of t h e a t o m s i n t h e a t m o s p h e r e a r e d e t e r m i n e d by s o l v i n g t h e d i f f u s i o n e q u a t i o n Oc = DV2C + ~D" V (cvu) + VdV c, 0"7

(4)

w h e r e t h e f i r s t t e r m on t h e r i g h t d e s c r i b e s t h e flow of a t o m s due t o t h e c o n c e n t r a t i o n g r a d i e n t , t h e s e c o n d r e p r e s e n t s d i f f u s i o n in t h e f i e l d of t h e e l a s t i c p o t e n t i a l g r a d i e n t , and t h e t h i r d r e p r e s e n t s t h e outflow o r inflow of a t o m s a s a r e s u l t of the m o t i o n of t h e d i s l o c a t i o n s ; D i s t h e d i f f u s i o n c o e f f i c i e n t of t h e i m p u r i t y atoms. F i g u r e 3 i l l u s t r a t e s t h e d~mamic f r i c t i o n a l f o r c e a s a f u n c t i o n of t h e d i s l o c a t i o n v e l o c i t y ( c u r v e 1) f o r a r e c t i l i n e a r d i s l o c a t i o n m o v i n g at a c o n s t a n t v e l o c i t y u n d e r s t e a d y - s t a t e c o n d i t i o n s ~ c / a t = 0 in c o o r d i n a t e s of r d / T ~ n a x - q / q m a x . H e r e r ~ a x i s t h e m a x i m u m f r i c t i o n a l f o r c e

Tn~ax = (/3 i s a n u m e r i c a l c o n s t a n t ) ;

ACo ~-a~-

q m a x c o r r e s p o n d s to t h e v a l u e of t h e p a r a m e t e r

(5) q f o r w h i c h Td = -r dm a x , t h i s

q u a n t i t y u s u a l l y b e i n g c l o s e to unity; q = Vd/Vc w h e r e v c = D / l c i s t h e m i g r a t i o n v e l o c i t y of t h e a t m o s p h e r e ; l c = A / k T i s t h e s t a t i c r a d i u s of t h e a t m o s p h e r e . C u r v e 2 in F i g . 3 i l l u s t r a t e s (in t h e c o o r d i n a t e s F / F m a x - - q / q m a x ) t h e v e l o c i t y d e p e n d e n c e of t h e f r i c t i o n a l f o r c e a s s o c i a t e d with t h e f o r m a t i o n of Snoek a t m o s p h e r e s a r o u n d t h e d i s l o c a t i o n s , a s d e r i v e d b y E s h e l b y [22]. T h e v e l o c i t y f u n c t i o n F r e a c h e s a m a x i m u m F m a x = 0.35 f o r q m a x = 4 / 3 . W e s h o u l d n o t e t h e f u n c t i o n a l s i m i l a r i t y b e t w e e n t h e two v e l o c i t y d e p e n d e n c e s of Td. H e n c e , if t h e d i s l o c a t i o n m o v e s a t a c o n s t a n t v e l o c i t y , f o r s t e a d y s t a t e m o t i o n t h e r e t a r d i n g f o r c e of t h e a t m o s p h e r e will b e

w h e r e F i s a f u n c t i o n s i m i l a r t o t h e E s h e l b y v e l o c i t y f u n c t i o n . F o r Vd/V c >> 1 t h i s h a s a n a s y m p t o t i c a p p r o x i m a t i o n F(q) = q-1 l n q / 3 J , ln'g = 0.577; h o w e v e r , if q << 1 we h a v e F(q) ~- q. A c h a r a c t e r i s t i c f e a t u r e of the v e l o c i t y d e p e n d e n c e of t h e f r i c t i o n a l f o r e e of t h e a t m o s p h e r e i s a r e d u c t i o n in s t r e s s ( r o u g h l y a s 1/Vd) when t h e v e l o c i t y of t h e d i s l o c a t i o n s t a r t s e x c e e d i n g t h e r a t e of m i g r a t i o n of t h e a t m o s p h e r e v c . T h i s r e d u c t i o n i s a r e s u l t of t h e f a e t t h a t , w i t h i n c r e a s i n g v e l o c i t y of t h e d i s l o c a t i o n , t h e v o l u m e of c r y s t a l s f r o m which t h e d i s s o l v e d a t o m s a r e a b l e t o m i g r a t e t o t h e d i s l o c a t i o n s d i m i n i s h e s , a s a r e s u l t of w h i c h t h e c o n c e n t r a t i o n of i m p u r i t y a t o m s c l o s e to t h e d i s l o c a t i o n f a l l s r a p i d l y . T h e e x t e n t of t h i s v o l u m e m a y b e a r b i t r a r i l y c a l l e d t h e d y n a m i c r a d i u s of t h e a t m o s p h e r e l d. It f o l l o w s f r o m Eq. (4) t h a t l d = l e ( v c / V d ) 1/2 [14]. W h e n Vd/V c > 1 t h e d y n a m i c r a d i u s i s s m a l l e r t h a n t h e s t a t i c r a d i u s , and f o r a c e r t a i n c r i t i c a l v e l o c i t y of t h e d i s l o c a t i o n it e q u a l s t h e l a t t i c e c o n s t a n t . T h i s m e a n s t h a t f o r t h e s e and a n y h i g h e r v e l o c i t i e s h a r d l y any a t m o s p h e r e i s f o r m e d a r o u n d t h e d i s l o c a t i o n s , i . e . , t h e d i s l o c a t i o n s w i l l m o v e a s f r e e e n t i t i e s (rd = 0). Thus if f o r s o m e r e a s o n t h e v e l o e i t y of a d i s l o c a t i o n r i s e s s h a r p l y and s u b s t a n t i a l l y e x c e e d s Ve, t h e a t m o s p h e r e a r o u n d i t w i l l v a n i s h , in o t h e r w o r d s t h e d i s l o c a t i o n w i l l be " d e t a c h e d " f r o m t h e a t m o s p h e r e . On the o t h e r hand, if a d i s l o c a t i o n without a n y a t m o s p h e r e s t a r t s t r a v e l l i n g m o r e s l o w l y , a n a t m o s p h e r e will d e v e l o p a r o u n d it, t h e m o r e d e n s e l y s o t h e l o w e r t h e v e l o c i t y of t h e d i s l o e a t i o n and t h e l o n g e r t h e s l o w i n g - d o w n t i m e . F o r low v a l u e s of Vd(Vd/V e << 1) t h e a t m o s p h e r e s w i l l h a v e a n a l m o s t e q u i l i b r i u m c o n c e n t r a t i o n of i m p u r i t y a t o m s c ,~ e0 e - u / k T . In t h i s e a s e t h e f r i c t i o n a l f o r c e a r i s e s f r o m t h e f a c t t h a t , d u r i n g the m o t i o n of t h e d i s l o c a t i o n , t h e c e n t e r of g r a v i t y of t h e a t m o s p h e r e d o e s not c o i n c i d e with t h e dislocation. In o u r s u b s e q u e n t a n a l y s i s of s p a s m o d i c d e f o r m a t i o n we s h a l l c o n f i n e a t t e n t i o n to t h e s t e a d y - s t a t e a p p r o x i m a t i o n (6) f o r t h e d y n a m i c f r i c t i o n a l f o r c e . W e m u s t v e r i f y t h e c o n d i t i o n s u n d e r which t h i s i s j u s t i f i e d . C l e a r l y t h e t i m e r e q u i r e d to r e a c h a s t e a d y s t a t e of m o t i o n t d = 12d/D o r t d = / c / V d a l l o w i n g f o r t h e l d - - v d r e l a t i o n s h i p . T h i s t i m e i s t h e s h o r t e r , the h i g h e r t h e v e l o c i t y of t h e d i s l o c a t i o n . S i n c e t h e v e l o c i t i e s of t h e d i s l o c a t i o n s a s s o c i a t e d with t h e P o r t e v i n - - L e C h a t e l i e r e f f e c t m a y v a r y o v e r a w i d e r a n g e , t h e l i r n i t i n g f a c t o r m a y b e t h e t i m e t c = 1 / 4 pD r e q u i r e d to f o r m a n e q u i l i b r i u m a t m o s p h e r e a r o u n d a s t a t i o m q r y d i s l o c a t i o n [23] (p i s the d i s l o c a t i o n d e n s i t y ) . T h e s e two q u a n t i t i e s s h o u l d b e s m a l l enough f o r a p p r o x i m a t i o n (6) to be v a l i d . T h e s t e a d y - s t a t e a p p r o x i m a t i o n f o r t h e r e t a r d i n g f o r c e of t h e a t m o s p h e r e

138

V,,

Vo

o,,5I- - ~e (~)

rJrT~ e/e~, KT

!

I

~/~

I

I

qo,

I

I

I

la

Fig. 3 Fig. 4 Fig. 3. D y n a m i c r e t a r d i n g f o r c e of t h e a t m o s p h e r e a s a f u n c t i o n of t h e v e l o city of the dislocations: 1) c o m p u t e r s o l u t i o n of Eq. (4) [14]; 2) E s h e l b y f u n c tion [22]. F i g . 4.

G r a p h i c a l s o l u t i o n of Eq. (13).

imposes no limitation on the basic results of the analysis, but it influences the quantitative relationships in that range of parameters within which it is applicable. Dynamic Equation. For analyzing the spasmodic deformations of types B and C we may use the dynamic quasihomogeneous approximation [24]. The nucleus of a Portevin--Le Chatelier band arising as the result of ~i fluctuation, constituting that part of the sample in which the dislocations have been "detached" f r o m the atmosphere, produces a fall in the load. The new s t r e s s is insufficient for the rapid motion of the dislocations, and atmospheres accordingly redevelop around these. For a net, band to develop, the s t r e s s level must be raised to at least its previous value. The net, band appears in front of the old (usually alongside it), since the part of the sample behind it has been strengthened more than the part in front. As a result of this, the bands, as it were, "leap" along the sample. Experiment [25] shows that the width of the bands B approximately equals the sample diameter d oand depends (even if slightly) on the temperature and the rate of deformation. The effective length of the sample is thus d0//0 times shorter than the length of the sample l 0. It is reasonable to assume that in a r e gion equal to the t'idth of the band deformation takes place homogeneously. If the rigidity of the test machine is finite, the rate of plastic deformation =

eo

K

'

(7)

H e r e e 0 is t h e d e f o r m a t i o n r a t e s p e c i f i e d by t h e m a c h i n e (allowing for l o c a l d e f o r m a t i o n ) ; K is t h e e f f e c t i v e m o d u l u s of t h e s a m p l e / t e s t m a c h i n e s y s t e m ; 1 / K = 1 / E + S 0 / I K ' , w h e r e l ~ do; S O is t h e s a m p l e c r o s s s e c t i o n ; K' i s t h e r i g i d i t y of t h e m a c h i n e ; E is t h e Y o u n g ' s m o d u l u s . S u b s t i t u t i n g (2) into (7) and a l l o w i n g f o r (1), we o b t a i n t h e d y n a m i c e q u a t i o n of t h e s a m p l e / t e s t m a c h i n e s y s t e m 9 80 --

x = ~0e

-~"

H(x--xi--Xd-{-',gln

)

hr

(8)

T h e d y n a m i c and d i f f u s i o n e q u a t i o n s (8) a n d (4) and a l s o t h e e q u a t i o n s f o r t h e d y n a m i c r e t a r d i n g f o r c e of t h e a t m o s p h e r e (3) and t h e v e l o c i t y of the d i s l o c a t i o n s va = - ~

(9)

Kbp

f o r m a c l o s e d s y s t e m of e q u a t i o n s w h i c h a l l o w s f o r t h e i n f l u e n c e of t h e d y n a m i c c h a r a c t e r i s t i c s m a c h i n e a n d the s t r u c t u r a l s t a t e of t h e m a t e r i a l on i t s d e f o r m a t i o n b e h a v i o r .

of t ~

C o n d i t i o n s f o r t h e A p p e a r a n c e of S p a s m o d i c D e f o r m a t i o n . In o r d e r to find p e r i o d i c s o l u t i o n s t o t h e s y s t e m of e q u a t i o n s (3), (4), (8), (9) we i n t r o d u c e c e r t a i n s i m p l i f i c a t i o n s . We a s s u m e t h a t t h e t o t a l y i e l d s t r e s s r m a y b e e x p r e s s e d in t h e f o r m of a s u m r = r + r ~ , w h e r e r = r * - - r i i s the m e a n s t r e s s , v a r y i n g s l o w l y with t i m e a n d d e f o r m a t i o n , r ~ i s t h e v a r i a b l e o s c i l l a t o r y s t r e s s I r ~ l << r . In c o n f o r m i t y with t h e e x p e r i m e n t s of [10] w e s h a l l c o n s i d e r t h a t t h e s t r u c t u r a l p a r a m e t e r s p, e0' and Ti v a r y in the u s u a l w a y

139

with d e f o r m a t i o n (as in t h e a b s e n c e of s p a s m o d i c d e f o r m a t i o n ) . We f u r t h e r a s s u m e t h a t the s t e a d y - s t a t e a p p r o x i m a t i o n (6) is v a l i d f o r Td. T h i s a s s u m p t i o n i s a v e r y s u b s t a n t i a l one, and it r e s t r i c t s t h e s u b s e quent r e s u l t s , s i n c e it f a i l s to d e t e r m i n e the e f f e c t of d e f o r m a t i o n r a t e and t e m p e r a t u r e on t h e a m p l i t u d e and p e r i o d of t h e l o a d o s c i l l a t i o n s c o m p l e t e l y . S u b j e c t to the f o r e g o i n g a s s u m p t i o n s , we r e p l a c e t h e s y s t e m of e q u a t i o n s (3), (4), (8), (9) by a s i n g l e d y n a m i c e q u a t i o n f o r t h e o s c i l l a t i n g s t r e s s ~-~ [24]

(10) K~o w h e r e ~.o0 = ( K ' / M ) ~/2 i s t h e c h a r a c t e r i s t i c f r e q u e n c y of t h e s a m p l e / m a c h i n e s y s t e m ; V i s t h e a c t i v a t i o n volume; V = --dH/dr*; q0 = g 0 / b 0 v c . E q u a t i o n (10) is t h e o s c i l l a t o r y e q u a t i o n with a f r i c t i o n a l f o r c e ( c u r l y b r a c k e t s ) n o n l i n e a r l y d e p e n d i n g on t h e v e l o c i t y . T h e f i r s t t e r m in t h e c u r l y b r a c k e t s d e s c r i b e s t h e f r i c t i o n a l f o r c e a s s o c i a t e d with t h e t h e r m a l l y - a c t i v a t e d d i s l o c a t i o n o v e r c o m i n g the b a r r i e r s with a s h o r t r a n g e of a c t i o n ; the s e c o n d t e r m d e s c r i b e s t h e d y n a m i c r e t a r d i n g f o r c e of t h e a t m o s p h e r e . F o r z e r o i n i t i a l c o n d i t i o n s T~ = 0 and ~'~ = 0 t h e e x c i t a t i o n of v i b r a t i o n s in t h e s y s t e m d e s c r i b e d by Eq. (10) is only p o s s i b l e u n d e r c e r t a i n c o n d i t i o n s , w h i c h d e p e n d on t h e i n t e r n a l p r o p e r t i e s of t h e s y s t e m . If t h e s y s t e m s a t i s f i e s t h e s e r e q u i r e m e n t s it will be s e l f - e x c i t i n g o r s e l f - o s c i l l a t o r y [26, 27]. In s u c h a s y s t e r n o s c i l l a t i o n s a r e g e n e r a t e d a s t h e r e s u l t of a r a n d o m d e v i a t i o n f r o m t h e s t e a d y s t a t e of m o t i o n . S i n c e a t t h e i n i t i a l i n s t a n t of t h e o s c i l l a t o r y m o t i o n t h e v a l u e of ~-~ i s s m a l l , we m a y l i m i t a t t e n t i o n to the f i r s t t e r m in § in t h e e x p a n s i o n f o r t h e f r i e t i o n a l f o r c e and i n s t e a d of (10) o b t a i n a l i n e a r e q u a t i o n for the oscillating stress "~~ + p~~ + o)~1:~ = O,

(1])

where

P =

K~o P0; Po = wT----~ + qoF' (qo); F'=

d_(.F

dqo"

T h e g e n e r a l s o l u t i o n to Eq. (10) m a y b e w r i t t e n in the f o r m

t (12)

~~ = B e 4o sin (~ot + a),

w h e r e B and ~ a r e the i n i t i a l c o n s t a n t s a s s o c i a t e d with r a n d o m d e v i a t i o n s of t h e s t r e s s e s the zero values;

to=---~

; r162

~ V 1 - - (,)

~~ or T~ from

.

F o r o s c i l l a t i o n s with a n i n c r e a s i n g a m p l i t u d e t o a p p e a r , it is c l e a r l y n e c e s s a r y t h a t t o > 0 a n d a l s o I%1 > 1 / ~ 0. In o r d e r to s a t i s f y t h e f i r s t c o n d i t i o n we m u s t h a v e P0 < 0, i . e~ kT - - qo F" (qo) > - -

(13)

T h i s e q u a t i o n m a y b e shown to be a n e c e s s a r y c o n d i t i o n f o r t h e d e v e l o p m e n t of P o r t e v i n - - L e C h a t e l i e r b a n d s . In o r d e r to s a t i s f y t h e e q u a t i o n t h e v e l o c i t y d e p e n d e n c e of t h e y i e l d s t r e s s m u s t h a v e a s e c t i o n with ~T//~E ( 0. T h e s e c o n d c o n d i t i o n It01 > 1 / ~ 0 m e a n s t h a t t h e e x c i t a t i o n t i m e s h o u l d b e g r e a t e r t h a n t h e p e r i o d of t h e n a t u r a l o s c i l l a t i o n s of t h e s y s t e m 1 / ~ 0 . T h i s c o n d i t i o n , which m a y a l s o be d e s c r i b e d in t h e f o r m ~0_fK )~ ] P0 [, {00T~nax i s the c o n d i t i o n f o r t h e d e v e l o p m e n t of j u m p s of t y p e s B and C. In t h e c a s e of t h e t y p i c a l v a l u e s K = 103 k g / r a m 2 , J0 = 10+3 s e c - 1 , IP01 = 10-2 we find t h a t t h e o n s e t of C j u m p s r e q u i r e s g0 > 10-2 s e c - 1 . H e n c e , on a l l o w i n g f o r the l o c a l i z a t i o n of t h e d e f o r m a t i o n , the " c a l c u l a t e d " d e f o r m a t i o n r a t e s h o u l d b e g r e a t e r

140

(14)

Fig. 5. R a n g e of e x i s t e n c e of the P o r t e v i n - - L e C h a t e l i e r effect i n c o o r d i n a t e s of In --1/T.

eros. t h a n 10 -4 s e c -1. F o r l o w e r v e l o c i t y j u m p s of the A type will o c c u r . of j u m p s to the o t h e r d e p e n d s on the t e m p e r a t u r e .

T h e r a t e of t r a n s i t i o n f r o m one t~q3e

T h e c h a r a c t e r of the e x c i t a t i o n (soft o r h a r d ) d e p e n d s on the v a l u e of to: the s h o r t e r the t i m e to, the h a r d e r is the e x c i t a t i o n . F o r s u f f i c i e n t l y high t e m p e r a t u r e s P0 ~ q0 F'(q0) ~ - - b p v c / ~ 0 so that K~o

to

(o2bpvc~n~'x

It follows f r o m t h i s e q u a t i o n that, as the t e m p e r a t u r e and r a t e of m i g r a t i o n of the a t m o s p h e r e Ve i n c r e a s e , t h e e x c i t a t i o n t i m e b e c o m e s s h o r t e r and s h o r t e r . T h u s according" to the e x p e r i m e n t a l data the a p p e a r a n c e of a type C s p a s m o d i c d e f o r m a t i o n r e q u i r e s a n a d e q u a t e r i g i d i t y of the m a c h i n e , a high t e m p e r a t u r e , and a high r a t e of d e f o r m a t i o n . Soft e x c i t a t i o n (low t e m p e r a t u r e s ) is c h a r a c t e r i z e d by a s m o o t h r i s e in the a m p l i t u d e of the o s c i l l a t i n g s t r e s s [17]. Let us c o n s i d e r the c o n s e q u e n c e s a r i s i n g f r o m c o n d i t i o n (13). F i g u r e 4 p r e s e n t s the g r a p h i c a l s o l u t i o n of Eqo (13), w h e r e F is the E s h e l b y f u n c t i o n [22]. T h e r a n g e of p a r a m e t e r s c o r r e s p o n d i n g to the e x i s t e n c e of t h e P o r t e v i n - - L e C h a t e l i e r effect (shaded r e g i o n ) is l i m i t e d by two c r i t i c a l v a l u e s of the p a r a m e t e r , q0(q01 < q0 < q02 and the c o n d i t i o n k T / V r ~ n a x < 0.137. T h i s l a t t e r e n a b l e s us to find the m i n i m u m c o n c e n t r a t i o n of i m p u r i t y a t o m s n e e d e d for the a p p e a r a n c e of the s p a s m o d i c d e f o r m a t i o n 1 [g2 w h e r e fl' = 0.22/3.

E s t i m a t e s show that c~aain a p p r o x i m a t e l y e q u a l s 10 -4 [24].

F u r t h e r , s i n c e q01 ~ q m a x = 4 / 3 , while q0 F'(q0) (for q0 >> 1) has a n a s y m p t o t i c a p p r o x i m a t i o n qo 1 lnq0 / 7 1 , TI = 4.84, the c o n d i t i o n q01 < q0 < q02 m a y be e x p r e s s e d in the f o r m 4 -3-

e0

J ~, Vr~'ax

<-~-~r ......

kr

(16) '

w h e r e k = In (q02/T1). F o r a c o n s t a n t d e f o r m a t i o n r a t e the l e f t - h a n d s i d e of i n e q u a l i t y (16) c h a r a c t e r i z e s the h i g h - t e m p e r z t u r e l i m i t to the e x i s t e n c e of the effect:

'(88 F o r the l o w - t e m p e r a t u r e l i m i t , a c c o r d i n g to (16) we have 9

eo

9

;k~

copD ,

(1 8)

w h e r e k = 0 . 7 - 5 . 1 as q0 v a r i e s f r o m 10 to 103 . U n d e r c o n s t a n t - t e m p e r a t u r e c o n d i t i o n s the r a t i o of (18) to (17) d e t e r m i n e s the r a n g e of d e f o r m a t i o n r a t e s w i t h i n which the effect e x i s t s (Fig. 5). F o r c h a r a c t e r i s t i c v a l u e s V = 100 a 3, ~ = a3/4, i e = 10 a , fl = 17, k = 3 a n d b = a (a is the l a t t i c e c o n s t a n t ) , the r a t i o of t h e m a x i m u m to the m i n i m u m v e l o c i t y is 1 . 5 - 104c0 . F o r a t y p i c a l c o n c e n t r a t i o n c o = 1%, the width of the band e q u a l s two o r d e r s of the v e l o c i t y v a l u e s . If the c o n c e n t r a t i o n c0 is l e s s t h a n the c r i t i c a l v a l u e (c o ~ 10-4), the band width equals zero, i . e . , t h e r e is no s p a s m o d i c d e f o r m a t i o n . 141

Conditions (17) and (18) a r e critical in the sense that a slight deviation f r o m these conditions causes the effect to vanish completely. Experiment [10] in fact shows that a r i s e of only 2~ in t e m p e r a t u r e close to the upper boundary leads to the complete disappearance of the jumps, although before this their amplitude had been s e v e r a l o r d e r s of magnitude above the sensitivity of the m e a s u r i n g s y s t e m . The condition r e g a r d i n g the critical c o m p r e s s i o n of the a t m o s p h e r e (by virtue of which the dimensions of the a t m o s p h e r e become s m a l l e r than the lattice constant) gives the following c r i t e r i o n for the upper limit to the existence of the effect [14] 9

/

lob\

e, <~ ~-~-) pD.

(19)

The ratio of" the criteria (18) and (19) equals k ~(a2/Icb)(V/~)Coor 2 9103c0 for the same values of the parameters as those used above. The criterion (18) will l i m i t the range of existence of the effect i f co < 10-3. For higher concentrations the upper l i m i t according to (19) should not depend on the value of co i f the remaining parameters such as V do not do so. For constant temperature the band width will be (l c /a) 2, i.e., as before, two orders of the deformation velocity. The temperature dependence of the critical velocities emin (17) and ernax (18) or (19) is determined by the temperature dependence of the diffusion mobility of the dissolved atoms (Fig. 5). However, experiment shows that the quantity emin usually depends on the temperature more strongly than ~max [6, 7, 12, 14]. For the velocity emin the slope of the straight line In emin - (1/T~ corresponds to the activation energy of the diffusion of the dissolved atoms [7, 14], while the slope of the line In amax --(1/T~ amounts to 0.4-0.6 of this quantity. It is usually assumed that this difference may be attributed to the nonequilibrium deformation vacancies, which start playing a substantial part at the relatively low temperature corresponding to the l i m i t ~max [2]. The diffusion mobility of the dissolved atoms D will then be proportional to cve-Um7kT where um is the migration energy of the vaeancies or vacancy/impurity atom complexes, Cv(e ) is the concentration of the deformation vacancies~ The fact that the nonequilibrium vacancies influence the onset of spasmodic deformation was clearly established in [11]. However, attempts at observing the influence of the deformation vacancies on the onset of the effect by direct experiment have so far been unsueeessful [10, 28]. It follows from the foregoing discussion that spasmodie deformation arises over a f a i r l y p~arrow range of dislocation velocities (Vd/Vc ~ 1-102). For the mobility of the dislocations at specified ~0 and T to fall within this range, a certain specific density of the mobile dislocations is essential. At the starting point of the strain diagrams, when the dislocation density is f a i r l y low, the dislocations move rapidly, and as regards velocity lie beyond the upper l i m i t of the effect. If, however, the dislocation density and the concentration of the deformation vacancies reach the values necessary for the dislocation velocity to fall into the critical range, oscillations of the load w i l l appear on the diagram, these being associated with the generation of localized Portevin--Le Chatelier zones of plastic deformation in the sample. Equation (1) describes the very earliest stages in the appearance of load oscillations. Steady-state oscillations with a steady amplitude and frequeney are described by the nonlinear Eq. (10). The amplitude and frequency of these relaxation oseillations were determined in [24] for the case in which the dynamic frictional force depended solely on the velocity.

LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

142

CITED

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14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 9 24. 25. 26. 27. 28.

H. Yoshinaga and S. Morozumi, Phil. Mag., 23, No. 186 (1971). B. J. Brindley and P. J. Worthington, Metals and Materials, 4, No. 8 (1970). P. C. McCormick, Acta Met., 19, No. 5 (1971). A. Wijler, Sehade van Westrum J., Scripta Met., 5, No. 10 (1971). S. R. Bodner and J. Barueh, J. Appl. Phys., 45, No. 5 (1972). W. N. Sharpe, J. Mech. Phys. Solids, 14 (1966). P. Penning, Acta Met., 20, No. 10 (1972). J . Guillot and J. Grilhe, Acta Met., 20, No. 2 (1972). J. D. Eshelby, Phil. Mag., 6, No. 68 (1961). R. Bullough and R. Newman, in: T h e r m a l l y Activated P r o c e s s e s in C r y s t a l s (V. L. Indenbom and A. N. Orlov editors) [Russian translation], Mir, Moscow (1973): G . A . Malygin, Phys. Status Sol., 15a, No. 1 (1973). A. Wijler, Schade van W e s t r u m J., Scripta Met., 5, No. 2 (1971). N . L . Kaidanovskii and S. t~. Khaikin, Zh. Tekh. Fiz~ 3, No. 1 (1933). J. Stocker, Nonlinear Vibrations [Russian translation], IL, Moscow (1952). P . J . Lloyd and P. J. Worthington, Phil. Mag., 24, No. 187 (1972).

143