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Reply to “comments on dislocation dynamics during rate changes”
ScTipta M E T A L L U R G I C A
Vol, 12, pp. 655~656, 1978 Printed in the U n i t e d States
P e r g a m o n Press,
Inc.
REPLY TO "COMMENTS ON D I S L O C A T I O N D Y N A M I C S D U R I N G RATE CHANGES"
B. W i e l k e and G. S c h o e c k I n s t i t u t f~r F e s t k S r p e r p h y s i k , U n i v e r s i t y V i e n n a B o l t z m a n n g a s s e 5, A - 1 0 9 0 Vienna, Austria (Received May 25, 1978)
In the p r e c e d i n g a r t i c l e (i) P. J. J a c k s o n d i s c u s s e s the t r a n s i e n t s in the s t r e s s - s t r a i n curve w h i c h o c c u r after a sudden strain rate change. F o l l o w i n g the ideas of Basinski, J a c k s o n and D u e s b e r y (2) (in the f o l l o w i n g BJD) he p r o p o s e s that t h e s e t r a n s i e n t s are c a u s e d by a c h a n g e in w o r k - h a r d e n i n g and he q u e s t i o n s the e x p l i c a t i o n that the t r a n s i e n t s result from a change in m o b i l e d i s l o c a t i o n d e n s i t y (3 - 8). Before e n t e r i n g into d i s c u s s i o n we w a n t to p o i n t out an a p p a r e n t m i s c o n c e p t i o n in J a c k s o n ' s arguments. He states that: "Changes in m o b i l e disl o c a t i o n d e n s i t y Pm w o u l d affect the rate at w h i c h d i s l o c a t i o n s a c c u m u l a t e and hence the w o r k - h a r d e n i n g rate." H o w e v e r the w o r k - h a r d e n i n g "rate" is not the inc r e a s e of stress T per time i n c r e m e n t but per glide i n c r e m e n t Aa = ApLb. As long as the m e a n free p a t h L of d i s l o c a t i o n s stays constant, the strain i n c r e m e n t Aa d e p e n d s o n l y on the a c c u m u l a t e d d e n s i t y of d i s l o c a t i o n s Ap w h i c h have m o v e d but not on the d e n s i t y Pm of s i m u l t a n e o u s l y m o v i n g ones. W i t h o u t e n t e r i n g into a l e n g t h y d i s c u s s i o n of the paper of BJD (2) we w a n t to p o i n t out that their i n t e r p r e t a t i o n is b a s e d on a m e r e l y p h e n o m e n o l o g i c a l des c r i p t i o n of the t r a n s i e n t s w h i c h leads to p h y s i c a l l y u n r e a l i s t i c c o n s e q u e n c e s , w h e n they are i n t e r p r e t e d u n c r i t i c a l l y : There seems to be a general a g r e e m e n t that the s t r a i n - r a t e a is c o n t r o l l e d by thermal a c t i v a t i o n and h e n c e can be r e p r e s e n t e d by = aoexp[-AG(T)/kT]
(i)
w h e r e AG is the free e n t h a l p y of activation, w h i c h d e p e n d s on the r e s o l v e d shear stress T. N o w BJD i n t r o d u c e two a d j u s t a b l e p a r a m e t e r s 1 , ~P , X = ~-o I - ~v-'" . ; -P L
~P 8 s = (~--L--). (3) P p Lp w i t h P = load, L = p l a s t i c e x t e n s i o n rate and 8 = w o r k - h a r d e n i n g rate in steady state, w h i c h up.t~ a g e o m e t r i c factor equals the Mormal w o r k - h a r d e n i n g c o e f f i c i e n t 8 = (~T/~a)A (Lp = const). W i t h equ. (I) and (2) we have e s s e n t i a l l y (2)
and
1 .~in ~ -i =
To
kT.~AG. -I =
_
=
kT v
w h e r e v is the so c a l l e d a c t i v a t i o n v o l u m e w h i c h g e n e r a l l y d e p e n d s on T. By c h o o s i n g X as a c o n s t a n t t h o u g h a d j u s t a b l e p a r a m e t e r a priori the a s s u m p t i o n is m a d e that v also stays c o n s t a n t d u r i n g the transient. This is h o w e v e r not v e r y res t r i c t i v e since it can be shown (6) that p h y s i c a l l y r e a s o n a b l e c h a n g e s in v are unable to e x p l a i n the e x p e r i m e n t a l l y o b s e r v e d transients. 655
656
REPLY TO COMMENTS ON DISLOCATION DYNAMICS
Vol.
12, No.
7
In the procedure taken by BJD the experimental curve was fitted (2): "... taking 8 from the slope of the hardening curve (BJD mean load vs time curve) after the sudden drop in stress and varying ~ until the best fit is obtained." Since the initial slope of the transient is independent of 8 (6), by choosing a corresponding ~ a good fit might be obtained covering just the early parts of the experimental curve using this set 8 and ~. The later parts of the observed transients deviate however increasingly and substantially from the theoretical curve. Therefore the claim of Jackson (i) to obtain a "better fit" between theory and experiment is misleading. Jackson uses this questionable "better fit" to claim also a posteriori justification for the assumption that during the transients the work-hardening coefficient changes. Though there is no way to decide, there are several serious objections to this assumption which make it rather unplausible: i) The selection of 8 taken from the slope of the work-hardening curve "after the sudden drop" is rather ambigous.
ip)
2) The value of 8 = 3T/~a is taken during the transient where a (or is not constant. Since T depends also on a (or i_) this value of 8 has no direct physical meaning, and without specifying Pthe dislocation velocity has no direct correlation to the physical work-hardening coefficient 8Q = (ST/~a)z which accounts for the accumulation of dislocations. As discussed above a a change in the density p_ of moving dislocations does not imply a different work-hardening coefficient. 3) The steady-state work-hardening 8= in stage II is independent of the strain rate a and a complicated model world have to be used to explain why it should change during the transients and then return to its original value. 4) Upon various successive up- and down-changes in strain rate the later part of the stress strain curve lies in the extrapolation of earlier parts. Therefore any increase in real work-hardening during down-changes would have to be compensated by a decrease or even negative work-hardening during up-changes. Such up-changes were not evaluated by BJD. There is however no physical realistic model why such decrease in work-hardening or even work-softening should occur during stage II. Though at the end of stage II and in stage III such effects are observed (9). In summary we may state that the fitting procedure of BJD using two adjustable parameters ~ and 8 may give some good agreement between measured and calculated curves, but only in the initial part of the transients. However the phenomenological parameter e must not be interpreted as work-hardening coefficient 8 = ~T/~a, and has no direct correlation to the underlying atomistic process. An uncritical interpretation leads to physical unrealistic consequences. References i. 2. 3. 4. 5. 6. 7. 8. 9.
P. Z. H. H. G. B. B. B. R. p.
J. Jackson, Scripta Met. 12, 655 (1978) S. Basinski, P. J. Jackson and M. S. Duesbery, Phil. Mag. 36, 255 (1977) Mecking and K. LOcke, Scripta Met. 4, 427 (1970) NeuhMuser,N.Himstedt and Ch.Schwink, Phys. Status Solidi (A) ~, 929 (1970) Schoeck and B. Wielke, Scripta Met. 10, 771 (1976) Wielke and G. Schoeck, Phys. Status Solidi (A), 38, 539 (1976) Wielke, Acta Phys. Aust. 48, 67 (1977) Wielke, Acta Met. 26, 103 (19Z8) Berner and H. Kronm~ller, Moderne Probleme der Metallphysik (Ed. A. Seeger) 59, Springer, Berlin-Heidelberg-New York (1965)
Vol, 12, pp. 655~656, 1978 Printed in the U n i t e d States
P e r g a m o n Press,
Inc.
REPLY TO "COMMENTS ON D I S L O C A T I O N D Y N A M I C S D U R I N G RATE CHANGES"
B. W i e l k e and G. S c h o e c k I n s t i t u t f~r F e s t k S r p e r p h y s i k , U n i v e r s i t y V i e n n a B o l t z m a n n g a s s e 5, A - 1 0 9 0 Vienna, Austria (Received May 25, 1978)
In the p r e c e d i n g a r t i c l e (i) P. J. J a c k s o n d i s c u s s e s the t r a n s i e n t s in the s t r e s s - s t r a i n curve w h i c h o c c u r after a sudden strain rate change. F o l l o w i n g the ideas of Basinski, J a c k s o n and D u e s b e r y (2) (in the f o l l o w i n g BJD) he p r o p o s e s that t h e s e t r a n s i e n t s are c a u s e d by a c h a n g e in w o r k - h a r d e n i n g and he q u e s t i o n s the e x p l i c a t i o n that the t r a n s i e n t s result from a change in m o b i l e d i s l o c a t i o n d e n s i t y (3 - 8). Before e n t e r i n g into d i s c u s s i o n we w a n t to p o i n t out an a p p a r e n t m i s c o n c e p t i o n in J a c k s o n ' s arguments. He states that: "Changes in m o b i l e disl o c a t i o n d e n s i t y Pm w o u l d affect the rate at w h i c h d i s l o c a t i o n s a c c u m u l a t e and hence the w o r k - h a r d e n i n g rate." H o w e v e r the w o r k - h a r d e n i n g "rate" is not the inc r e a s e of stress T per time i n c r e m e n t but per glide i n c r e m e n t Aa = ApLb. As long as the m e a n free p a t h L of d i s l o c a t i o n s stays constant, the strain i n c r e m e n t Aa d e p e n d s o n l y on the a c c u m u l a t e d d e n s i t y of d i s l o c a t i o n s Ap w h i c h have m o v e d but not on the d e n s i t y Pm of s i m u l t a n e o u s l y m o v i n g ones. W i t h o u t e n t e r i n g into a l e n g t h y d i s c u s s i o n of the paper of BJD (2) we w a n t to p o i n t out that their i n t e r p r e t a t i o n is b a s e d on a m e r e l y p h e n o m e n o l o g i c a l des c r i p t i o n of the t r a n s i e n t s w h i c h leads to p h y s i c a l l y u n r e a l i s t i c c o n s e q u e n c e s , w h e n they are i n t e r p r e t e d u n c r i t i c a l l y : There seems to be a general a g r e e m e n t that the s t r a i n - r a t e a is c o n t r o l l e d by thermal a c t i v a t i o n and h e n c e can be r e p r e s e n t e d by = aoexp[-AG(T)/kT]
(i)
w h e r e AG is the free e n t h a l p y of activation, w h i c h d e p e n d s on the r e s o l v e d shear stress T. N o w BJD i n t r o d u c e two a d j u s t a b l e p a r a m e t e r s 1 , ~P , X = ~-o I - ~v-'" . ; -P L
~P 8 s = (~--L--). (3) P p Lp w i t h P = load, L = p l a s t i c e x t e n s i o n rate and 8 = w o r k - h a r d e n i n g rate in steady state, w h i c h up.t~ a g e o m e t r i c factor equals the Mormal w o r k - h a r d e n i n g c o e f f i c i e n t 8 = (~T/~a)A (Lp = const). W i t h equ. (I) and (2) we have e s s e n t i a l l y (2)
and
1 .~in ~ -i =
To
kT.~AG. -I =
_
=
kT v
w h e r e v is the so c a l l e d a c t i v a t i o n v o l u m e w h i c h g e n e r a l l y d e p e n d s on T. By c h o o s i n g X as a c o n s t a n t t h o u g h a d j u s t a b l e p a r a m e t e r a priori the a s s u m p t i o n is m a d e that v also stays c o n s t a n t d u r i n g the transient. This is h o w e v e r not v e r y res t r i c t i v e since it can be shown (6) that p h y s i c a l l y r e a s o n a b l e c h a n g e s in v are unable to e x p l a i n the e x p e r i m e n t a l l y o b s e r v e d transients. 655
656
REPLY TO COMMENTS ON DISLOCATION DYNAMICS
Vol.
12, No.
7
In the procedure taken by BJD the experimental curve was fitted (2): "... taking 8 from the slope of the hardening curve (BJD mean load vs time curve) after the sudden drop in stress and varying ~ until the best fit is obtained." Since the initial slope of the transient is independent of 8 (6), by choosing a corresponding ~ a good fit might be obtained covering just the early parts of the experimental curve using this set 8 and ~. The later parts of the observed transients deviate however increasingly and substantially from the theoretical curve. Therefore the claim of Jackson (i) to obtain a "better fit" between theory and experiment is misleading. Jackson uses this questionable "better fit" to claim also a posteriori justification for the assumption that during the transients the work-hardening coefficient changes. Though there is no way to decide, there are several serious objections to this assumption which make it rather unplausible: i) The selection of 8 taken from the slope of the work-hardening curve "after the sudden drop" is rather ambigous.
ip)
2) The value of 8 = 3T/~a is taken during the transient where a (or is not constant. Since T depends also on a (or i_) this value of 8 has no direct physical meaning, and without specifying Pthe dislocation velocity has no direct correlation to the physical work-hardening coefficient 8Q = (ST/~a)z which accounts for the accumulation of dislocations. As discussed above a a change in the density p_ of moving dislocations does not imply a different work-hardening coefficient. 3) The steady-state work-hardening 8= in stage II is independent of the strain rate a and a complicated model world have to be used to explain why it should change during the transients and then return to its original value. 4) Upon various successive up- and down-changes in strain rate the later part of the stress strain curve lies in the extrapolation of earlier parts. Therefore any increase in real work-hardening during down-changes would have to be compensated by a decrease or even negative work-hardening during up-changes. Such up-changes were not evaluated by BJD. There is however no physical realistic model why such decrease in work-hardening or even work-softening should occur during stage II. Though at the end of stage II and in stage III such effects are observed (9). In summary we may state that the fitting procedure of BJD using two adjustable parameters ~ and 8 may give some good agreement between measured and calculated curves, but only in the initial part of the transients. However the phenomenological parameter e must not be interpreted as work-hardening coefficient 8 = ~T/~a, and has no direct correlation to the underlying atomistic process. An uncritical interpretation leads to physical unrealistic consequences. References i. 2. 3. 4. 5. 6. 7. 8. 9.
P. Z. H. H. G. B. B. B. R. p.
J. Jackson, Scripta Met. 12, 655 (1978) S. Basinski, P. J. Jackson and M. S. Duesbery, Phil. Mag. 36, 255 (1977) Mecking and K. LOcke, Scripta Met. 4, 427 (1970) NeuhMuser,N.Himstedt and Ch.Schwink, Phys. Status Solidi (A) ~, 929 (1970) Schoeck and B. Wielke, Scripta Met. 10, 771 (1976) Wielke and G. Schoeck, Phys. Status Solidi (A), 38, 539 (1976) Wielke, Acta Phys. Aust. 48, 67 (1977) Wielke, Acta Met. 26, 103 (19Z8) Berner and H. Kronm~ller, Moderne Probleme der Metallphysik (Ed. A. Seeger) 59, Springer, Berlin-Heidelberg-New York (1965)