DISLOCATION DRAG DUE TO RANDOMLY DISTRIBUTED POINT OBSTACLES

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DISLOCATION DRAG DUE TO RANDOMLY DISTRIBUTED POINT OBSTACLES

J. Schlipf

To cite this version:

J. Schlipf. DISLOCATION DRAG DUE TO RANDOMLY DISTRIBUTED POINT OBSTACLES.

Journal de Physique Colloques, 1985, 46 (C10), pp.C10-215-C10-218. �10.1051/jphyscol:19851048�.

�jpa-00225432�

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DISLOCATION DRAG DUE T O RANDOMLY DISTRIBUTED POINT OBSTACLES

J. SCHLIPF

Institut für Allgemeine Metallkunde und Metallphysik, Techn.

Hochschule. 0-5100 Aachen, F.R.G.

A b s t r a c t

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A mode1 i s presented f o r 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 motion Fn concentrated a l l o y s . It takes i n t o account t h e e x t r a f r i c t i o n e x e r t e d by t h e weaker obstacles when a s t r o n g e r o b s t a c l e i s overcome. I t p r e d i c t s an i n t e r n a 1 f r i c t i o n behavior which i s S t r o n g l y dependent on temperature, frequency, concentration, and s t r a i n amplitude.

1

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INTRODUCTION

The problem o f t h e r m a l l y a c t i v a t e d motion o f d i s l o c a t i o n s through a random a r r a y of p o i n t obstacles has been t r e a t e d n u m e r i c a l l y by several authors / 1 / . On t h e o t h e r hand, an approximate a n a l y t i c a l s o l u t i o n has been given f o r t h e case o f small con- c e n t r a t i o n s o f o b s t a c l e s /2/. I n o b t a i n i n g t h i s s o l u t i o n i t was shown t h a t even i n t h e presence o f a spectrum o f o b s t a c l e s t r e n g t h s d i s l o c a t i o n motion i s governed by a unique a c t i v a t i o n energy AHM. I n t h e f o l l o w i n g t h i s p r i n c i p l e o f unique a c t i v a t i o n i s used t o t r e a t t h e 'case o f concentrated s o l i d s o l u t i o n s , where c l u s t e r s o f s o l u t e s form obstacles o f v a r i o u s strengths. The r e s u l t s a r e then a p p l i e d t o d i s l o c a t i o n damping i n these a l l o y s .

II - STATISTICAL THEORY OF POINT OBSTACLES

As w i l l be shown elsewhere, t h e p r o b a b i l i t y o f f i n d i n g an o b s t a c l e o f s t r e n g t h

= nFo along a d i s l o c a t i o n l i n e i n a concentrated s o l i d s o l u t i o n i s approximately

L?ven by 2

c, = 4 c ( ~ / b ) ( 2 d ) - ' / ~ e x p ( - n /2m) ( 1 )

where b i s t h e Burgers v e c t o r , w t h e w i d t h o f t h e obstacle, m = en 2 ? t h e q u a d r a t i c average o f o b s t a c l e s t r e n g t h , and c t h e c o n c e n t r a t i o n o f solutes. For any c l a s s n o f obstacles t h e r e i s a s e t o f k i n e t i c equations which describe t h e steady motion o f t h e d i s l o c a t i o n w i t h r e s p e c t t o these obstacles /2/. Sumning up t h e c o n t r i b u t i o n s o f each c l a s s o f obstacles one o b t a i n s t h r e e c h a r a c t e r i s t i c q u a n t i t i e s :

( i ) the r e s t o r i n g f o r c e due t o t h e obstacles

Ca

=

-

^{JMcn F}^; ^dn

( i i ) t h e c r i t i c a l r e s o l v e d shear s t r e s s (CRSS)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19851048

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( i i i ) t h e t o t a l d r a g c o e f f i c i e n t due t o t h e obstacles Ëm = ( l / b )J cn F;/un*-dn 2

M

Here

r

i s t h e l i n e t e n s i o n o f t h e d i s l o c a t i o n and un* t h e forward jump frequency a t an o b s t a c l e o f s t r e n g t h n. The a s t e r i s k * denotes t h e c o n f i g u r a t i o n from which t h e t h e r m a l l y a c t i v a t e d jump i s made. M represents t h e s t r e n g t h o f those obstacles f o r which t h e unique a c t i v a t i o n energy j u s t equals t h e t o t a l i n t e r a c t i o n energy MUo.

Obstacles w i t h n < M w i l l c o n t r i b u t e o n l y l i t t l e t o t h e dragging.

From Eqs. ( 2 ) and ( 3 ) M and y a r e o b t a i n e d as f u n c t i o n s o f a. An approximate c a l - c u l a t i o n y i e l d s

m = 2cfi, M = alfi; Y = a21al (5a,b,c)

where al < 1 and ci2 > 1 a r e numerical f a c t o r s o f o r d e r 1.

III

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THE DRAGGING FORCE

With t h e d i s t r i b u t i o n o f o b s t a c l e s t r e n g t h s given by ( 1 ) any o b s t a c l e o f a given strength, n = n i Say, i s surrounded by a cloud o f weaker obstacles w i t h n < n i . I n an a c t i v a t i o n event a t n l t h e d i s l o c a t i o n sweeps o u t t h e a c t i v a t i o n area. I t t h e r e - f o r e has t o overcome a l 1 t h e weaker obstacles o f t h e c l o u d b e f o r e r e a c h i n g t h e saddle p o i n t o f t h e o b s t a c l e n l (Fig.1). This r e s u l t s i n an increased f r i c t i o n , Bnl, f e l t by t h e d i s l o c a t i o n . Denoting t h e c o n t r i b u t i o n o f obstacles o f s t r e n g t h n by B(n) we can c a l c u l a t e Bnl, t h e p a r t i a l drag c o e f f i c i e n t , from

n l

En, = J B(n)dn. ( 6 )

M

Considering t h e a c t i v a t i o n event as a s t o c h a s t i c process /3/ we may w r i t e f o r t h e jump frequency:

w,^ = v R ~ / ( 8 n / 2 ~ c n ) 2 + 1

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Ën/2BCnl exp(-nHM/kT) (7 1

where vR i s t h e a t t a c k frequency according t o r a t e theory, Bc? = nBcl i s a c r i t i c a l damping constant, determined by t h e c u r v a t u r e o f t h e i n t e r a c t ~ o n p o t e n t i a l a t t h e saddle p o i n t . L e t us consider t h e l i m i t i n g case En >> 2Bcn:

w,^ ^RIV ~ ( B ~ ~ / Ë ~ ) ~ X ~ ( - A H ~ / ~ T ) . (8)

Combining Eqs. ( 4 ) , ( 6 ) , and (8) and i n t r o d u c i n g HM = MUo, we have

d&,/dn = (cn F,* Ën/b vR Bcn)exp(MUo/kT) 2 ( 9 )

and upon i n t e g r a t i on:

As T goes t o zero,&, increases i n d e f i n i t e l y , so t h a t a r e g i o n o f En > Bcn i s always a t t a i ned.

I V

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INTERNAL FRICTION

Under t h e a c t i o n o f t h e above dragging f o r c e s t h e equation o f motion o f a d i s l o c a - t i o n f o r a p e r i o d i c a l l y a p p l i e d s t r e s s a = oo e x p ( i w t ) may be w r i t t e n :

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o f t h e form

2 2

A = A. LI T / ( ~ + u T ) (12)

w i t h A,

AL^

~ b ~ / 1 2 r ; T = B w ~ Z / ~ r (13a,b)

A and L represent d e n s i t y and loop length, r e s p e c t i v e l y , o f t h e d i s l o c a t i o n s par- t i c i p a t i n g i n t h e process. Due t o t h e double exponential o f 1/T i n Ëw t h e peaks as a f u n c t i o n o f temperature become v e r y narrow. They a r e a l s o s t r o n g l y dependent on s t r a i n amplitude. Examples a r e given i n Fig.2. For a d i r e c t comparison w i t h e x p e r i - ments exact numerical c a l c u l a t i o n s a r e probably necessary. However, extremely narrow peaks have indeed been r e p o r t e d i n t h e l i t e r a t u r e /5/.

REFERENCES

/1/ R. Labusch, G. Grange, K. Ahearn, and P. Haasen, i n : Rate Processes i n P l a s t i c '

Deformation of M a t e r i a l s , Amer. Soc. Met.

,

Metals Park Ohio ( 1 975) 26.

/2/ J. S c h l i p f , phys. s t a t . s o l . ( a ) 74 (1982) 529.

/3/ P. Stichaner, Di plomarbeit, Techn. Hochschule Aachen (1 973) ; R.D. Isaac, Thesi s, Univ. o f I l l i n o i s a t Urbana-Champain (1 977).

/4/ D. Lenz and K. Lücke, Proc. ICIFUAS-5, Springer-Verlag, B e r l i n (1975) 48.

/5/ M. I s e k i , M. Kiowa, M. Hi rabayashi, Proc. ICIFUAS-6, U n i v e r s i t y o f Tokyo Press (1977) 659; Q.P. Kong, K. Lücke, and G. Sokolowski

,

J. Physique, C o l l . C5, (1981) 271.

minor obstacles (n <nl)

Fig. 1

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Thermally a c t i v a t e d jump a t an o b s t a c l e o f s t r e n g t h n l surrounded by weaker obstacles. I n t r a v e r s i n g t h e shaded a c t i v a t i o n area t h e d i s l o c a t i o n experiences t h e e x t r a f r i c t i o n due t o t h e minor obstacles.

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Temperature I K 1

Fig. 2

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T y p i c a l i n t e r n a 1 f r i c t i o n peaks f o r a 10%-alloy, a t t h r e e s t r a i n amplitudes and a frequency o f 10 kHz.