STUDY OF DISLOCATION - POINT DEFECTS INTERACTION IN MgO SINGLE CRYSTALS BY INTERNAL FRICTION

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STUDY OF DISLOCATION - POINT DEFECTS INTERACTION IN MgO SINGLE CRYSTALS BY

INTERNAL FRICTION

M. Gabbay, Gilbert Fantozzi

To cite this version:

M. Gabbay, Gilbert Fantozzi. STUDY OF DISLOCATION - POINT DEFECTS INTERACTION

IN MgO SINGLE CRYSTALS BY INTERNAL FRICTION. Journal de Physique Colloques, 1981, 42

(C3), pp.C3-31-C3-41. �10.1051/jphyscol:1981303�. �jpa-00220694�

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Colloque C3, supplément au n°6, Tome 42, juin 1981 page C3-31

STUDY OF DISLOCATION - POINT DEFECTS INTERACTION IN MgO SINGLE CRYSTALS BY INTERNAL FRICTION

M. Gabbay and G. Fantozzi

Groupe d'Etudes de Métallurgie Physique et de Physique des matériaux, I.N.S.A. Lyon, Bâtiment 502, 20, Avenue A. Einstein, 69621 Villeurbanne Cedex, France

Résumé.- Après déformation plastique, on observe une restauration du Frotte- ment Intérieur (F.I.) lors des recuits, due au piégeage des dislocations par des défauts ponctuels. Cette arrivée des défauts ponctuels sur les disloca"- tions provoque l'apparition d'un phénomène de désancrage thermpmécanique sous l'effet de la contrainte de sollicitation. Nous avons étudié ce phénomène de désancrage responsable du F.I. AJJ dépendant de l'amplitude. Pour cela, nous avons déterminé la variation de AJJ avec l'amplitude de vibration £ soit en fonction de la température de mesure, soit en fonction de la température de recuit. Nous avons tenté d'interpréter nos résultats expérimentaux en fonc- tion des divers modèles de désancrage (modèles de Granato-Lvicke et de Teuto- nico, Granato et Lucke). Notre analyse montre qu'il est nécessaire de tenir compte de l'activation thermique pour rendre compte correctement de nos ré- sultats et nous obtenons ainsi une valeur de l'énergie d'interaction dislo- cation-défauts ponctuels de l'ordre de 0,4 eV.

A b s t r a c t . - After p l a s t i c deformation, an I n t e r n a l F r i c t i o n ( I . F . ) recovery i s observed during annealing, due to the pinning of d i s l o c a t i o n s by point de- f e c t s . This pinning induces thermomechanical breakaway under s t r e s s of v i b r a - t i o n . We have studied t h i s breakaway phenomenon responsible of amplitude de- pendent I . F . AH. We have determined the v a r i a t i o n of Ag versus s t r a i n ampli- tude e e i t h e r as a function of measurement temperature or as a function of annealing temperature. We have t r i e d to i n t e r p r e t our experimental r e s u l t s according to various breakaway models (model of Granato-Lucke and model of Teutonico, Granato, Lucke). Our analysis shows t h a t thermal a c t i v a t i o n has to be taken i n t o account and so we have obtained a value for i n t e r a c t i o n energy between d i s l o c a t i o n - point defects of about 0.4 eV.

1 . I n t r o d u c t i o n . - I n p r e v i o u s p a p e r s / 1 , 2 / , we h a v e a l r e a d y p r e s e n t e d some a s p e c t s o f t h e b e h a v i o u r o f m o b i l e d i s l o c a t i o n s i n MgO s i n g l e c r y s t a l s s t u d i e d by I n t e r n a l F r i c t i o n ( I . F . ) . Among t h e main r e s u l t s o f t h i s s t u d y , i t i s u s e f u l t o p o i n t o u t t h e h i g h s e n s i t i v i t y o f I . F .

( d u e t o d i s l o c a t i o n s ) t o a n n e a l i n g w h i c h b e g i n s t o b e o p e r a t i v e j u s t a b o v e Room T e m p e r a t u r e ( R . T . ) - d i s l o c a t i o n s a r e i n t r o d u c e d b y p l a s t i c d e f o r m a t i o n a t R . T . - .

A f t e r h e a t i n g a t 670 K, d i s l o c a t i o n d a m p i n g i s c o n s i d e r a b l y r e - d u c e d ; we know t h a t , i n t h i s a n n e a l i n g t e m p e r a t u r e r a n g e , t h e d i s l o c a - t i o n n e t w o r k i s n o t m o d i f i e d . S o , we h a v e n a t u r a l l y s u g g e s t e d t h a t t h i s I . F . r e c o v e r y c o n s e c u t i v e t o a n n e a l i n g i s d u e t o d i s l o c a t i o n p i n - n i n g by p o i n t d e f e c t s . M i g r a t i o n e n e r g y E o f t h e s e p o i n t d e f e c t s h a s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981303

C3-32 JOURNAL DE PHYSIQUE

been d e t e r m i n e d by a n a l y s i s o f a n n e a l i n g k i n e t i c s ; a l o n g t h e most p a r t o f t h e a n n e a l i n g t e m F e r a t u r e r a n g e , t h e v a l u e f o r Em remains between

1.5 eV and 1.8 eV. A s 1 . 5 6 eV i s t h e m i g r a t i o n e n e r g y f o r c a t i o n va- cancy / 3 / s o , we have c o n f i r m e d t h a t c a t i o n vacancy i s t h e mobile en- t i t y which makes e a s i e r t h e m i g r a t i o n o f many o t h e r s p o i n t d e f e c t s s u c h a s vacancy p a i r s o r i m p u r i t y vacancy complexes. I t i s o f i n t e r e s t now, t o d e t e r m i n e t h e b i n d i n g energy between d i s l o c a t i o n s and p i n n i n g p o i n t d e f e c t s . So, we have f i r s t l y r e c o r d e d a m p l i t u d e dependent I . P . v e r s u s s t r a i n a m p l i t u d e . Then we have a t t e m p t e d t o a p p l y t o t h e s e ex- p e r i m e n t a l r e s u l t s , some c l a s s i c a l models o f thermomechanical break- away.

2 . E x p e r i m e n t a l d e t a i l s and r e s u l t s . - MgO s i n g l e c r y s t a l s a r e c l e a v e d a l o n g (100) f a c e s (dimensions : 40 x 4 x 2 mm3). They a r e p l a s t i c a l l y deformed i n compression a l o n g t h e l o n g e s t a x i s o f 2 % . They a r e e x c i t e d e l e c t r o s t a t i c a l l y i n t h e i r fundamental f l e x u r a l mode o f v i b r a t i o n a t 10 kHz. F u r t h e r d e t a i l s c o n c e r n i n g p r e p a r a t i o n , t h e r m a l t r e a t m e n t , p l a s - t i c d e f o r m a t i o n o f specimens and I . F . measurements a r e p r e s e n t e d e l s e - where /4/. Amplitude dependent I . F . AH i s o b t a i n e d from t h e t o t a l de- crement A by s u b t r a c t i n g t h e a m p l i t u d e i n d e p e n d e n t decrement A I . S t r a i n a m p l i t u d e E due t o f l e x u r a l v i b r a t i o n i s non uniform a l o n g t h e l e n g t h and t h e t h i c k n e s s of specimens; co i s c a l c u l a t e d a s t h e maximum v a l u e o f s t r a i n a m p l i t u d e i n t h e specimen.

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F i g . 1 . - A f t e r p l a s t i c d e f o r m a t i o n a t 293 K, v a r i a t i o n s o f AH v e r s u s E~ a t v a r i o u s t e m p e r a t u r e s : 1 : 77 K, 2 : 121 K, 3 : 145 K, 4 : 195 K, 5 : 227 K.

F i g u r e 1 shows a t y p i c a l s e t o f c u r v e s o f a m p l i t u d e dependent I.F. AH p l o t t e d a g a i n s t s t r a i n a m p l i t u d e co f o r v a r i o u s measurement t e m p e r a t u r e s r a n g i n g from 77 K t o 2 9 3 K . These c u r v e s a r e o b t a i n e d immediately a f t e r p l a s t i c d e f o r m a t i o n . We can n o t i c e t h a t f o r a g i v e n s t r a i n , t h e magnitude o f AH i n c r e a s e s w i t h i n c r e a s i n g t e m p e r a t u r e , s u g g e s t i n g t h a t more d i s l o c a t i o n s a r e depinned a t h i g h e r t e m p e r a t u r e s . Three s i m i l a r s e t s o f c u r v e s a r e o b t a i n e d f o r d i f f e r e n t a n n e a l i n g s t a - t e s o f t h e specimen. (Ta = 4 2 2 K t 5 4 0 K , 5 8 9 K ) .

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g e n e r a l l e v e l o f AH d e c r e a s e s w h i l e a n n e a l i n g t e m p e r a t u r e i n c r e a s e s . F i g u r e s 2 , 3 , 4 show t h e r e s u l t s c o r r e s p o n d i n g t o t h e above mentioned a n n e a l i n g s t a t e s .

F i g . 2.- A f t e r a n n e a l i n g a t 422 K, v a r i a t i o n s o f LH v e r s u s Eo a t v a r i o u s tempera- t u r e s . 1 : 77 K , 2 : 117 K , 3 : 149 K , 4 : 182 K , 5 : 296 K , 6 : 348 K , 7 : 378 K 8 : 388 K , 9 : 421 K .

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Fig. 3.- A f t e r a n n e a l i n g a t 540 K, v a r i a t i o n s of AH v e r s u s E, a t v a r i o u s tempera- t u r e s . 1 : 77 K, 2 : 148 K, 3 : 291 K, 4 : 361 K, 5 : 380 K, 6 : 412 K, 7 : 473 K 8 : 5 0 3 K, 9 : 540 K.

Fig. 4.- A f t e r anneal'ing a t 589 K, v a r i a t i s n s of AH v e r s u s ^Eo at v a r i o u s tempera- t u r e s . 1 : 77 K, 2 : 111 K, 3 : 136 K, 4 : 207 K, 5 : 291 K, 6 : 411 K , 7 : 498 K, 8 : 269 K, 9 : 589K.

3. A n a l y s i s o f t h e r e s u l t s and d i s c u s s i o n . - Many t h e o r e t i c a l models have been developed i n o r d e r t o i n t e r p r e t d i s l o c a t i o n breakaway bher- m a l l y a c t i v a t e d o r n o t ; most o f them have been reviewed by P e r e z e t a l . /5/. We have used some o f t h e s e models from which i t i s e x p e c t e d t o deduce t h e i n t e r a c t i o n energy between d i s l o c a t i o n s and p i n n i n g p o i n t d e f e c t s .

F i r s t l y , t h e c l a s s i c a L Granato-Liicke / 6 / model h a s been used f o r one s e t o f AH ( E ~ ) c u r v e s o b t a i n e d a f t e r a n n e a l i n g a t 589 K .

Amplitude dependent I . F . AH i s g i v e n by t h e e x p r e s s i o n : 3

A H = A L ~ A1 0

where A i s t h e d e n s i t y o f d i s l o c a t i o n f r e e t o v i b r a t e . LN i s t h e d i s l o c a t i o n l e n g t h between u n b r e a k a b l e nodes.

LC i s t h e d i s l o c a t i o n l e n g t h between a d j a c e n t p i n n i n g p o i n t s . A1 and A2 a r e c o n s t a n t s i n v o l v i n g g e o m e t r i c and o r i e n t a t i o n p a r a m e t e r s

and t h e p i n n i n g f o r c e between p o i n t d e f e c t and d i s l o c a t i o n . E x p r e s s i o n (1) shows t h a t through t h e e x p o n e n t i a l dependence of

AH on LC any f a c t o r which c a u s e s L c t o d e c r e a s e a l s o c a u s e s AH t o d e c r e a s e . LC i s g i v e n by:

where WM i s t h e d i s l o c a t i o n - p o i n t d e f e c t i n t e r a c t i o n e n e r g y

Co i s t h e a v e r a g e c o n c e n t r a t i o n o f p o i n t d e f e c t s i n t h e l a t t i c e . The r e l a t i o n s (1) and ( 2 ) p r o v i d e t h e c o n n e c t i n g l i n k between d i s l o c a t i o n damping and i n t e r a c t i o n e n e r g y . I f t h i s a n a l y s i s i s v a l i d , a p l o t of I n v e r s u s w i l l b e i n t h e form o f a s t r a i g h t l i n e ,

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v e r s u s ;i; s h o u l d b e ¹

LC LC

a l s o i n t h e form o f a s t r a i g h t l i n e from which s l o p e , a v a l u e of t h e i n t e r a c t i o n e n e r g y WM w i l l b e deduced.

A s we can s e e o f f i g u r e 5 , t h e G.L. model do n o t f i t v e r y w e l l o u r r e s u l t s . The G.L. p l o t s f a l l i n t o s t r a i g h t l i n e s o n l y a t t h e be- g i n n i n g o f t h e - 1 r a n g e . N e v e r t h e l e s s , t h e s l o p e s o f t h e s e l i n e s have

&o A2

been c a l c u l a t e d and t h e f i g u r e 6 shows t h e p l o t o f ( I n

-

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1

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from which we have o b t a i n e d 0.035 e V f o r t h e i n t e r a c t i o n e n e r g y WM.

The Cranato-Liicke model which does n o t t a k e i n t o a c c o u n t t h e r m a l a c t i v a t i o n i s n o t v e r y w e l l v e r i f i e d by o u r r e s u l t s and l e a d s t o a v e r y low v a l u e o f i n t e r a c t i o n e n e r g y .

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Fig. 5.- A f t e r a n n e a l i n g a t 589 K, Granato-~Gcke p l o t s , a c c o r d i n g t o r e l a t i o n ( I ) a t v a r i o u s temperatures.1 : 291 K, 2 : 411 K , 3 : 500 K, 4 : 519 K, 5 : 589 K.

Fig. 6.- Logarithmic p l o t s a c c o r d i n g t o r e l a t i o n (2) i n o r d e r t o o b t a i n i n t e r a c t i o n energy.

Then, we have used models i n which t h e r m a l a c t i v a t i o n i s t a k e n i n t o a c c o u n t . The most p a r t o f them a r e b a s e d on t h e r e s u l t s o f Teuto- n i c ~ e t a l . /7/. c o n c e r n i n g thermomechanical breakaway o f a d i s l o c a t i o n p i n n e d by a s i n g l e p i n n i n g p o i n t . A c t i v a t i o n e n e r g i e s f o r d e p i n n i n g and r e p i n n i n g a r e d e t e r m i n e d from t h e v a l u e o f t h e p o t e n t i a l e n e r g y o f t h e d i s l o c a t i o n - p i n n i n g p o i n t system.

S a u l and Bauer /8/ have o b t a i n e d f o r a m p l i t u d e dependent I . P . AH t h e f o l l o w i n g e x p r e s s i o n :

0

[

^M

]

AH = A exp

- --

where a i s t h e r e s o l v e d s h e a r stress a c t i n g on a d i s l o c a t i o n . v i s t h e a c t i v a t i o n volume.

3 2

A. i s t h e r e l a x a t i o n s t r e n g t h f a c t o r ( p r o p o r t i o n a l t o A LN / ^LC).

For e a r l y s t a t e s o f d e p i n n i n g i . e . f o r t h e c a s e where t h e f r a c - t i o n o f unpinned network l e n g t h f ( t ) i s s m a l l ( f ( t ) << 1) and f o r h i g h t e m o e r a t u r e ( u v < < kT) t h e r e l a t i o n ( 3 ) becomes:

T h e r e f o r e , AH(T) i s e x p e c t e d t o v a r y n e a r l y e x p o n e n t i a l l y w i t h 1, T f o r low v a l u e s o f A H .

Such an e x p e c t e d r e s u l t c a n b e a l s o deduced from t h e model pro- posed by F r i e d e l / 9 / . I n t h i s model, t h e a c t i v a t i o n e n e r g y f o r b r e a k - away i s g i v e n by:

where d i s t h e w i d t h of p o i n t d e f e c t . The a m p l i t u d e dependent I . F . A, i s ;

T h i s r e l a t i o n i s n o t v a l i d i n t h e low s t r e s s e s r a n g e b e c a u s e d i s l o c a t i o n r e p i n n i n g i s n o t t a k e n i n t o a c c o u n t . I t i s supposed a l s o t h a t t h e d e p i n n i n g o f t h e f i r s t p i n n i n g p o i n t o c c u r s a t a f r e q u e n c y much l o w e i t h a n t h e d e p i n n i n g f r e q u e n c y of t h e f o l l o w i n g p o i n t s .

F i g u r e 7 shows t h a t . t h e p l o t o f I n AH v e r s u s

T

1 d o e s n o t f a l l i n t o s t r a i g h t l i n e s a l o n g t h e t e m p e r a t u r e r a n g e . Here a g a i n , t h e s e two s i m i l a r models a r e n o t v e r i f i e d c o m p l e t e l y . N e v e r t h e l e s s , t h e i n - t e r a c t i o n e n e r g y deduced from t h i s p l o t i s 0.1 eV; t h i s v a l u e s e e m s t o b e t o o low c o m p a r a t i v e l y t o t h e t e m p e r a t u r e r a n g e i n which 1,F. i s

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a m p l i t u d e dependent.

A.. to4

l o p a t v a r i o u s F i k . 7.- According t o r e l a t i o n ( 4 ) , l o g a r i t h m i c p l o t s of AH v e r s u s

-

-5 -5

s t r a l n a m p l i t u d e s . 1 : Eo = 2 . 5 x 2 : Eo = 3.5 x 10

,

3 : E0 = 4.5 x 10

.

For t h i s r e a s o n , we have t r i e d t h e model proposed by Liicke e t a l . /lo/. These a u t h o r s have used t h e r e s u l t s o f T e u t o n i c o e t a l . / 6 / f o r t h e d e t e r m i n a t i o n of a m p l i t u d e dependent I . F . AH. The f o l l o w i n g r e l a t i o n g i v e s t h e v a r i a t i o n o f A H a s a f u n c t i o n o f t h e p a r a m e t e r T d e t e r m i n e d by t h e a p p l i e d s t r e s s , t e m p e r a t u r e and f r e q u e n c y .

L o

where ^{T =}

-

⁼

---

⁰

l o ⁰ T1rL

L i s t h e a v e r a g e l e n g t h of d i s l o c a t i o n l o o p .

lo i s t h e minimum l o o p l e n g t h under which breakaway c a n n o t o c c u r f o r o = o a t a g i v e n t e m p e r a t u r e .

0

' T ~ , L i s g i v e n by:

v1 i s t h e a t t e m p t f r e q u e n c y . v i s t h e v i b r a t i o n f r e q u e n c y .

"0

IJ % - i s t h e mechanical d e p i n n i n g s t r e s s . b 2 ~

uo i s t h e a c t i v a t i o n e n e r g y f o r breakaway.

For a g i v e n v a l u e o f AH, t h e c o r r e s p o n d i n g s t r a i n a m p l i t u d e E~

c a n be e x p r e s s e d a s a f u n c t i o n o f t e m p e r a t u r e a c c o r d i n g t o t h e f o l l o - wing r e l a t i o n /11/:

w i t h

Y i s t h e Young's modulus.

X i s a c o n s t a n t w h i c h d e p e n d s o n AH (1 = 0.36 when A H i s maxi- mum). I n o u r c a s e X h a s been e s t i m a t e d a t 0.2 f o r AH = 0 . 7

So t h e p l o t o f eo v e r s u s T ~ s h o u l d g i v e a s t r a i g h t l i n e from / ~ which t h e t e m p e r a t u r e Tc i s deduced and c o n s e q u e n t l y t h e a c t i v a t i o n energy Uo = 2 0 kTc. F u r t h e r m o r e , t h e s l o p e g i v e s t h e a c t i v a t i o n volume b 2 L.

P i g . 8.- According t o r e l a t i o n (9), p l o t of c0 v e r s u s T''~, a t c o n s t a n t A =

H 0 . 7 x and f o r v a r i o u s a n n e a l i n g t e m p e r a t u r e . 1 : 293 K , 2 : 422 K , 3 : 540 K 4 : 589 K.

c3-40 JOURNAL DE PHYSIQUE

F i g u r e 8 shows p l o t s o f c0 ( T ~ ' ~ ) fior d i f f e r e n t a n n e a l i n g s t a t e s . Each o f t h e s e p l o t s p r e s e n t s p a r t i a l l i n e a r i t y i n t h e low t e m p e r a t u r e r a n g e (T < 200 K ) . The h i g h e r t h e a n n e a l i n g t e m p e r a t u r e i s t h e s t r o n g e r t h e t e m p e r a t u r e dependence o f co i s . However t h e t e m p e r a t u r e T remains

C

c o n s t a n t ( s e e f i g u r e 8 , Tc c a n be d e f i n e d a s an i n t e r c e p t w i t h tempera- t u r e a x i s of t h e e x t r a p o l a t i o n o f t h e l i n e a r p a r t s o f t h e E (!P1I2)

0

p l o t s ) .

T h i s remark s u g g e s t s t h a t , i n t h e low t e m p e r a t u r e r a n g e , d i s l o c a - t i o n s i n t e r a c t w i t h p i n n i n g p o i n t d e f e c t s w i t h t h e same a c t i v a t i o n e n e r - gy Uo w e have o b t a i n e d Tc = 225 K and Uo = 0.4 e V . T h i s v a l u e c o u l d c o r r e s p o n d t o t h e i n t e r a c t i o n o f c a t i o n v a c a n c i e s o r impurity-vacancy complexes a s s u g g e s t e d by t h e v a l u e p r e v i o u s l y o b t a i n e d f o r m i g r a t i o n e n e r g y /3/ and by t h e c a l c u l a t i o n o f P u l s /14/.

Then, t h e a c t i v a t i o n volume v c a n be deduced from t h e s l o p e o f t h e l i n e a r p a r t s f o r e a c h a n n e a l i n g s t a t e s . We have o b t a i n e d :

Along t h e h i g h t e m p e r a t u r e r a n g e (T > 200 K ) t h e s t r a i n a m p l i t u d e

E~ (needed t o o b t a i n a g i v e n v a l u e o f A H ) d e c r e a s e s more and more slow- l y w h i l e t e m p e r a t u r e i n c r e a s e s . Such a b e h a v i o u r s u g g e s t s t h a t a n o t h e r i n t e r a c t i o n mechanism w i t h h i g h e r a c t i v a t i o n e n e r g y i s o p e r a t i v e .

'The r e s u l t s o b t a i n e d f o r a c t i v a t i o n volume a l l o w t o g e t an i d e a o f t h e r e d u c t i o n r a t i o o f a v e r a g e l e n g t h L o f f r e e d i s l o c a t i o n from a g i v e n a n n e a l i n g s t a t e t o t h e n e x t one.

W e have o b s e r v e d a r e d u c t i o n much more i m p o r t a n t t h a n t h e one deduced from t h e measurements of a m p l i t u d e i n d e p e n d e n t I . F . A I s i n c e A, i s p r o p o r t i o n a l t o L~ / 6 / . Such a d i f f e r e n c e between t h e s e two me- t h o d s c o u l d b e i n t e r p r e t e d e i t h e r by an a s y m m e t r i c a l p i n n i n g o f d i s l o - c a t i o n o r by t h e f a c t t h a t o n l y a s m a l l f r a c t i o n o f d i s l o c a t i o n s i s r e s p o n s i b l e f o r A H .

We have used v a r i o u s t h e o r e t i c a l models and e a c h o f them l e a d s t o a d i f f e r e n t v a l u e f o r a c t i v a t i o n e n e r g y . Such a d i s p e r s i o n o f t h e r e s u l t s had been a l r e a d y p o i n t e d o u t by De B a t i s t /12/. The v a l u e s o b t a i n e d from t h e G . L . model and t h e ~ a u . 1 Bauer model seems t o b e unrea- l i s t i c . The a n a l y s i s deduced from t h e T.G.L. model g i v e s a more p l a u - s i b l e v a l u e of i n t e r a c t i o n e n e r g y , c o m p a r a t i v e l y t o t h e t e m p e r a t u r e r a n g e i n which d e p i n n i n g i s o p e r a t i v e . I n o r d e r t o g e t a more a c c u r a t e a n a l y s i s , a t h e o r e t i c a l s t u d y o f thermomechanical breakaway n e e d s t o b e

u n d e r t a k e n . I n t h i s s t u d y , w e i n t e n d t o u s e a model p r e v i o u s l y deve- l o p e d by V i n c e n t e t a l . /13/. I . F . d u e t o d i s l o c a t i o n breakaway w i l l b e c a l c u l a t e d by n u m e r i c a l methods w i t h o u t a p p r o x i m a t i o n s a n d t h e r e - s u l t s w i l l b e compared t o t h e p r e s e n t e x p e r i m e n t a l r e s u l t s .

R e f e r e n c e s

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