ACCURATE DETERMINATION OF PARAMETERS OF INTERNAL FRICTION ASSOCIATED WITH GRAIN BOUNDARY

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ACCURATE DETERMINATION OF PARAMETERS OF INTERNAL FRICTION ASSOCIATED WITH

GRAIN BOUNDARY

Jiang Zi-Ying, Cai Wei-Ping

To cite this version:

Jiang Zi-Ying, Cai Wei-Ping. ACCURATE DETERMINATION OF PARAMETERS OF INTERNAL FRICTION ASSOCIATED WITH GRAIN BOUNDARY. Journal de Physique Colloques, 1985, 46 (C10), pp.C10-379-C10-382. �10.1051/jphyscol:19851085�. �jpa-00225470�

JOURNAL DE PHYSIQUE

C o l l o q u e C10, s u p p l e m e n t a u n 0 1 2 , Tome 4 6 , d e c e m b r e 1 9 8 5 page C10-379

ACCURATE DETERMINATION OF PARAMETERS OF INTERNAL FRICTION ASSOCIATED WITH GRAIN BOUNDARY

JIANG ZI-YING and CAI WEI-PING*

y o r t h e a s t U n i v e r s i t y o f T e c h n o l o g y , S h e n y a n g , L i a o n i n g , C h i n a Wuhan I r o n and S t e e l U n i v e r s i t y , Wuhan, H u b e i , C h i n a

A h s t r a c t - An e q u a t i o n t o exprpss g r a i n - h o u n d a r y i n t e r n a l f r i c t i o n was developed and e s t a h l i s h e r l . We use t h i s e q u a t i o n t o f i t e x p e r i m e n t a l c u r v e s

of i n t e r n a l f r i c t i o n a s s o c i a t e d w i t h grain-houndary m o t i o n hy c o m p u t a t i o n a l t e c h n i q u e s . According t o t h i s method, a s e r i e s o f micro-parameters f o r g r a i n - b o u n d a r y i n t e r n a l f r i c t i o n was o b t a i n e d f r o m a s i n g l e e x p e r i m e n t a l curve. T h i s e q u a t i o n and computer program was checked w i t h T.S. Ke's work on p u r e A1 / I / , showing t h a t t h e r e s u l t i s s a t i s f a c t o r y .

I - INTRODUCTION

The i n t e r n a l f r i c t i o n 0-I of a s t a n d a r d s o l i d i s g i v e n b y

where w i s t h e a n g u l a r f r e q u e n c y o f v i b r a t i o n , T i s t h e r e l a x a t i o n t i m e , AM i s t h e r e l a x a t i o n s t r e n g t h , and ,$ i s t h e l o s s angle. The r e l a t i o n between T and t h e a h s o l u t e t e m p e r a t u r e T i s g i v e n by T = T exp H/RT i n which H i s t h e

a c t i v a t i o n energy, T t h e f r e q u e n c y f a c t o r , 8nd R i s t h e gas c o n s t a n t . F o r most m a t e r i a l s , however, ?he r e l a x a t i o n t i m e T i s n o t s i n g l e , h u t d i s t r i b u t e d a c c o r d i n g t o some f u n c t i o n . So i t i s necessary t o e s t a b l i s h a f a v o r a b l e e q u a t i o n w h i c h w i l l i n v o l v e n o t o n l y t h e parameters H, A M , T ~ , h u t a l s o t h e parameter 6 which i s a measure o f t h e d i s t r i b u t i o n w i d t h o f r e l a x a t i o n t i m e T. Ry u s u a l methods a g r e a t amount o f d a t a i n c l u d i n g b o t h i n t e r n a l f r i c t i o n and o t h e r e x p e r i m e n t a l r e l a x a t i o n d a t a a r e r e q u i r e d t o d e t e r m i n e d H. AM, T , and B(T). Rut B(T) i s i m p o s s i h l e t o o b t a i n o n l y by i n t e r n a l f r i c t i o n data. I n a d d i t i o n , t h e procedure i s t e d i o u s . I n t h i s paper, t h e a u t h o r s make an a t t e m p t t o d e v e l o p an e q u a t i o n o f g r a i n boundary i n t e r n a l f r i c t i o n w h i c h i n v o l v e s t h e parameters above, and t h o s e parameters may be o b t a i n e d from a s i n g l e grain-boundary i n t e r n a l f r i c t i o n e x p e r i m e n t a l c u r v e u s i n g a p r o p e r computer program.

I 1 - EXPERIMFNTAL DATA PROCESSING

( A ) S u b t r a c t i o n o f t h e background. Reference /3/ mentioned t h a t s i n g l e c r y s t a l A1 i n t e r n a l f r i c t i o n as a f u n c t i o n o f t e m p e r a t u r e i s w e l l g i v e n hy

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19851085

JOURNAL DE PHYSIQUE

oh-1 ^{= A} exp (-BIT) ( 1

when T i s more t h a n 430 K, where t h e parameters A and B a r e t a k e n f r o m e x p e r i m e n t a l data. Reference 141 a1 so i n d i c a t e d t h a t many p o l y c r y s t a l l i n e m a t e r i a l s a r e s u b j e c t t o e q u a t i o n ( I ) , b u t t h e r e a r e g e n e r a l l y d e v i a t i o n s on t h e

l o w temperature s i d e o f t h e g r a i n boundary i n t e r n a l f r i c t i o n peak. Our measure- ment o f A1-Re a l l o y s i s i n agreement w i t h /4/. (See f i g . 1.) We suppose t h a t t h i s d e v i a t i o n may be caused by two reasons: One i s t h e e x t e r n a l f r i c t i o n which

r i s e s f r o m t h e o s c i l l a t i g m o t i o n o f i n e r t i a elements i n n o t t o o h i g h vacuum

a

c o n d i t i o n s (e.g. 5 x 10- T o r r ) . The o t h e r i s i n t e r n a l f r i c t i o n which a r i s e s f r o * sample i t s e l f , a l t h o u g h we d o n o t know t h e r e a l mechanism o f it. We t h i n k t h a t b o t h o f t h e s e may be regarded as temperature i n d e ~ e n d e n t . L e t us c a l l t h i s t o t a l damping c a p a c i t y t h e 0 - l . Then t h e apparent background i n t e r n a l f r i c t i o n 0-- 1 o b t a i n e d from measurement may h e expressed as 0-:= 0 - t + 0-k , where O-b i s t h e 1 t r u e background i n t e r n a l f r i c t i o n which i s temperature dependent as i n e q u a t i o n

(1). Thus we o b t a i n

We choose s e v e r a l p o i n t s a t pos;tions f a r beyond t h e peak area o f t h e e x p e r i m e n t a l g r a i n boundary i n t e r n a l f r i c t i o n p l o t (see p o i n t s 1-6 i n Fig. 2). By 1 in e a r r e a r e s s i o n

n 1 1 2

R = 1 r l n (Oci - 0-,) - ( I n A

-

i =l (3

where R i s t h e c o r r e l a t i o n c o e f f i c i e n t . Although 0 - I i s unknown, i t s v a l u e e x i s t s

1 1

i n t h e r e g i o n between O and minimum of O-c, namely

9:

s e c t i o n " me?hod, u n t i l R reaches a minimum. At t h e same t i m e , t h e c o n s t a n t s A and B a r e obtained. From t h i s t h i n k i n g , t h e corre.sponding computer program i s worked o u t .

I n Fig. 2, c u r v e A i s t h e e x p e r i m e n t a l g r a i n boundary i n t e r n a l f r i c t i o n vs.

temperature f o r A1-0.06% RE a l l o y s . Curve B i s t h e c o r r e s p o n d i n g background c u r v e o b t a i n e d by t h e computer. The s e m i l o g a r i t h m i c p l o t o f background i n t e r n a l

f r i c t i o n i n c u r v e R versus 1/T i s shown i n Fig. 1. These d a t a a r e i n d i c a t e d b y

I 1

.

c u r v e i s o b t a i n e d by t h e s u b t r a c t i o n o f background c u r v e B as shown i n F i g . 3, and t h e c u r v e i s normalized.

( B ) D e t e r m i n a t i o n o f micro-parameters. It i s known t h a t t h e d i s t r i b u t i o n of

T r e s u l t s from t h e d i s t r i b u t i o n o f T~ o r H, o r b o t h o f t h e s e 151. For simp1 i f i c a t i o n , we assume t h a t i ) An i n t e r n a l parameter 5 i s temperature2 independent, and obeys a Gaussian d i s t r i b u t i o n g(D) = lfi exp [- ( D l h ) 1,

where D = 5 - ⁶ , 5, i s t h e most probabl e v a l u e o f 6 , and A i s a measurement o f d i s t r i b u t i o n w i a t h o f 5 . i i ) I n T and H a r e b o t h d i r e c t l y p r o p o r t i o n a l

t o 5, i.e. I n T = PC; H = q ~ . P a h q a r e t h e p r o p o r t i o n a l i t y c o n s t a n t s . Thus, t h e d i s t r i b u t i o 8 f u n c t i o n $(Z) o f T i s w r i t t e n as

+(z) = l l g f i exp ~ - ( z / B ) ~ J ( 4 )

where Z = I n (TIT,), T,,, i s t h e most probahl e v a l u e of T: 6 = ( 6 + B ~ / T ) and 6 = ph, 6H = qh/R. 6, 6 and B~ a r e t h e measurements o f ?he d i s t r i b u t i o n w i d t h g o f I n T, i n r 0 and H r e ~ p e c t i v e l y . F o r convenience, 0 - i i s expressed i n terms o f T. We assume t h a t T,,, = T x exp (H,/RT), where T 1s t h e v a l u e o f r0 which corresponds t o T~ , H,,, ?Re most p r o b a b l e v a l u e o?"H. We o b t a i n

" w2 1 1

~ 1 2 6 l e- sech CHm/R (T - T) + BW) dW

- ^m

t g u = 0 - I = - m

b ca

A 1 1 -1 (5

1 + - J e - ' {I + exp 2rHm/R (T - T) + B W ~ } dW

J;;

--

where T,,, i s t h e t e m p e r a t u r e when W T = 1 and W = Z/B. E q u a t i o n ( 5 ) shows t h a t t h e peak temperature Tp = Tm o n l y when 8 ⁼ 0, and g e n e r a l l y Tp # Tm.

According t o e q u a t i o n (51, t h e c o r r s p o n d i n g computer program i s worked o u t w i t h

" p a c e - l e n g t h a c c e l e r a t i n g " method and t h e parameters can be determined. (The d e t a i l s o f t h e procedure a r e r e p o r t e d elsewhere.) The r e l i a b i l i t y o f e q u a t i o n ( 5 ) and t h e computer-program i s checked a g a i n s t Ke's e a r l y work on p u r e (99.99%) A1 / 6 / . Table 1 shows t h e s e parameters; t h e v a l u e o b t a i n e d i n o u r work i s l i s t e d i n t h e f i r s t row, and t h e v a l u e s o b t a i n e d f r o m t h e r e f e r e n c e concerned a r e l i s t e d i n t h e second row.

Table 1

Parameter Hm B~ eH 8(Tp) Tp Tm A t g ( T p )

O u r w o r k 1.39eV 2.62 2.94 3.27 285.2C 292C 0.41 0.0797 8.35

I n r e f e - 1.40eV / / 3.30 285C / 0.49 0.080 /

rence

From t a b l e 1 i t can be seen t h a t b o t h a c c o r d w e l l . The advantage of o u r work i s t h a t o n l y a s i n g l e e x p e r i m e n t a l grain-boundary re1 a x a t i o n c u r v e i s o b t a i n e d , 1 i ke A1 a l l o y s , n i n e parameters can be determined a t once, and frequency change by a f a c t o r o f t w o o r t h r e e i s n o t needed. Using t h i s method, we i n v e s t i g a t e d t h e g r a i n - b o u n d a r y i n t e r n a l f r i c t i o n o f A1-RE a l l o y s , and o b t a i n e d s a t i s f a c t o r y r e s u l t s ( s e e o t h e r paper o f a u t h o r s i n t h e s e conference p r o c e e d i n g s ) .

Fig: 1 Curves of

ln& vs I/T f o r ~ l - O.O6%RE, 450C 2hr annealed.

Points a r e taken f a r from peak area.

1 HZ experimental points

o 0.5 Hz experimen- t a l points

value a f t e r sub- t r a c t i o n of ~ 6 1 .

C10-382 JOURNAL DE PHYSIQUE

Fig. 2 I n t e r n a l f r i c t i o n c u r v e P i g . 3 Normalj.zation c u r v e s of a s s o c i a t e w i t h g r a i n - g r a i n boundary i n t e r n a l boundary r e l a x a t i o n f r i c t i o n f o r A1-0.0663~~:

A : Experimental curve.

B: Back-ground c u r v e o b t a i n e d w i t h com- p u t e r p r o c e s s i n g .

R e f f e r e n c e

/I / T.S.Ke PhyS. Rev. 71 (1 947) 533.

/2/ A.S.Nowick, B.S.Berry. IBM. J.Res. Develop, 5(1961) 297. 712.

/3/ D.H.Niblet a n d J.Wilks. Adv. Phys. g(1960) 1.

/4/ G.Schoeck. E.B3-dgni and J.Shyne 12(1964) 1466.

/5/ A.S .Nowick, B.S .Berry. A n e l a s t i c R e l a x a t i o n i n C r y s t a l l i n e S o l i d s Academic P r e s s (1 972).

/ 6 / T.S.Ke Phys. Rev. 71(1947) 533.