DISLOCATION RELAXATION IN A RANDOM ARRAY OF SOLUTES

(1)

HAL Id: jpa-00221089

https://hal.archives-ouvertes.fr/jpa-00221089

Submitted on 1 Jan 1981

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

DISLOCATION RELAXATION IN A RANDOM

ARRAY OF SOLUTES

I. Ritchie, A. Atrens, C. So, K. Sprungmann

To cite this version:

I. Ritchie, A. Atrens, C. So, K. Sprungmann.

DISLOCATION RELAXATION IN A

(2)

JOURNAL DE PHYSIQUE

CoZZoque C5, supple'ment au nolO, Tome 42, octobre 1981 page C5-319

D I S L O C A T I O N R E L A X A T I O N I N A RANDOPZ ARRAY O F SOLUTES I.G. Ritchie, A. ~trens*, C.B. So and K.W. Sprungmann

MateriaZs Science Branch, Atomic Energy of Canada Limited, Whiteshe22 NucZear Research EstabZishment, Pinawa, Manitoba ROE 1L0, Canada

" B B C Brown, Boveri and Compar~y, Limited, CH-5405 Baden-DattwiZ, SuitzerZand

Abstract.- An amplitude-dependent internal friction peak occurs at 1 475 K in

well-annealed, pure a-Zr after slight deformation. Qualitatively, the results agree with a model for the motion of zig-zag dislocations in an array of obstacles. Comparison of theoretical and experimental results yields estimates of the effective activation enthalpy and activation volume of the process involved.

1. Introduction.- Recently, there have been advances in the understanding of high- temperature, solid-solution hardening (SSH) by dynamic simulation using computer

techniques. For example, Schwarz and L.abusch(') have employed such a technique to

c~lculate the SSH component of the flow stress (of), while Schwarz (2) has used the

same programs to calculate amplitude (€)-dependent internal friction (A). The results

reveal a simple quantitative correspondence between A ( € ) and of. These studies are

providing new insight into the nature of strain-aging effects and the athermal plateau in af vs. temperature (T) curves for solution-strengthened alloys.

In a-Zr, anomalous mechanical behaviour (particularly creep) and the athermal

plateau in the o VS. T curve is attributed to the strong interactions between dis-

locations and interstitial impurities (usually oxygen) ( 3 ) . In the same T range,

peaks of A vs. E with some of the characteristic properties of a thermally assisted

unpinning process have been It is shown in this paper that these

peaks can be generated in well-annealed specimens of comparatively high purity a-Zr after a very slight deformation at room temperature. Consequently, we can compare the results with calculations based on a very simple model proposed by Schlipf and

~chindlma~r(~-~), which is valid for the motion of zig-zag dislocations in a random

array of solute obstacles in the low concentration limit.

2. Material.- Measurements of A in flexure (frequency, f 4 Hz) were carried out on

specimens of a nominally 5N pure Zr. The interstitial impurity concentrations (vg/g)

of the as-received (cold-rolled strip) material were found to be H < 5, N = 6, and 0 in the range 17 to 23. Specimens were used which had been annealed in a vacuum

furnace at 1075 K (final grain size 0.7 mm) and then slightly deformed in bending

at room temperature. Even though precautions against the pick-up of 0 were taken,

after annealing and testing analysis revealed some pick-up of both 0 and H. The

final concentrations (pg/g) were H = 14

+

2 and 0 = 30. The equipment and measure- ( 9 )

ment techniques employed have been described in detail elsewhere

.

(3)

C5-320 JOURNAt DE PHYSIQUE

3. R e s u l t s . - T y p i c a l c u r v e s of A v s . T ( E c o n s t . ) and A v s . E (T c o n s t . ) a r e shown i n f i g u r e s 1 and 2 , r e s p e c t i v e l y . It must b e p o i n t e d o u t , however, t h a t i d e n t i c a l r e s u l t s were n o t o b t a i n e d on e v e r y specimen t e s t e d , presumably b e c a u s e t h e a t t e m p t e d " v e r y s l i g h t d e f o r m a t i o n " d i d n o t a l w a y s s u c c e e d . N e v e r t h e l e s s , once t h e y had been o b t a i n e d w i t h a g i v e n specimen, t h e r e s u l t s were s u r p r i s i n g l y s t a b l e and r e p r o d u c i b l e on h e a t i n g and c o o l i n g , p r o v i d e d t h a t t h e specimen was n o t h e a t e d above a b o u t 510 K. T h i s i s i n agreement w i t h t h e work of ~ t r e n s ' ~ ) who showed t h a t a s i m i l a r peak d i s -

a p p e a r e d a f t e r h e a t i n g f o r 20 m i n u t e s a t 500 K. 3.0 _I _I _I ₁ _I _l /'l. 480K

/

.' 2.0

-

'... .. _-...

-

497 K 4 0 0 450 500 550 . .-Ll. 1. TEMPERA1URE.K O h 4 5 6 7

NOMINAL SURFACE STRAIN AMPLITUDE

F i g . 1 : I n t e r n a l f r i c t i o n a s a f u n c t i o n F i g . 2 : I n t e r n a l f r i c t i o n a s a f u n c t i o n of t e m p e r a t u r e a t v a r i o u s c o n s t a n t - s t r a i n o f nominal s t r a i n a m p l i t u d e a t v a r i o u s a m p l i t u d e s . The v a l u e s of E i n d i c a t e d c o n s t a n t t e m p e r a t u r e s . The s t r a i n ampli- a r e a p p r o x i m a t e l y 5 x

l o 7

t i m e s t h e s t r a i n t u d e i s 5 X 1 0 - ~ t i m e s t h e nominal s t r a i n a m p l i t u d e f o r R = 2/ ( 3 ~ ) . a m p l i t u d e f o r R = 2 / ( 3 n ) .

F i g u r e 1 c a n be d e s c r i b e d a s a f a m i l y of p e a k s of a l m o s t c o n s t a n t h e i g h t , w i t h a common high-T t a i l , and i n c r e a s i n g b r o a d e n i n g on t h e low-T s i d e , a s c i n c r e a s e s . The r e s u l t s i n f i g u r e 2 c a n be d e s c r i b e d by a A v s . E peak which p a s s e s t h r o u g h t h e "window" o f e x p e r i m e n t a l l y i n v e s t i g a t e d E , from h i g h E t o l o w , & , a s T i n c r e a s e s . The s h a p e s and b e h a v i o u r o f t h e p e a k s i n f i g u r e s 1 and 2 a r e s i m i l a r t o t h o s e p r e d i c t e d t h e o r e t i c a l l y f o r t h e r m a l l y a s s i s t e d u n p i n n i n g o f Because o f t h e p r e p a r a t i o n of t h e s p e c i m e n s , t h e f r e s h d i s l o c a t i o n s s h o u l d b e "zig-zagged1' t h r o u g h a random a r r a y o f i n t e r s t i t i a l s o l u t e s . 4. T h e o r e t i c a l ~ o d e l ( ~ - ~ ) . - I t i s assumed t h a t t h e f r e s h l y c r e a t e d network l e n g t h s c a n be t r e a t e d by t h e v i b r a t i n g - s t r i n g model b u t , b e c a u s e o f t h e i r i n t e r a c t i o n s w i t h t h e random a r r a y of o b s t a c l e s , e a c h n e t w o r k l e n g t h (L) w i l l t a k e on a z i g - z a g con- f i g u r a t i o n c h a r a c t e r i z e d by a n a m p l i t u d e (2) and a l e n g t h ( R ) a s shown i n f i g u r e 3. A t t r a c t i v e and r e p u l s i v e o b s t a c l e s a r e supposed t o b e p r e s e n t i n e q u a l numbers, and t h e " i d e a l i z e d " b a s i c s t e p i n t h e d i s l o c a t i o n m o t i o n i s t a k e n t o b e t h e jump from one z i g - z a g c o n f i g u r a t i o n , An Bn AnC1, i n t o a n e q u i v a l e n t o n e , An Cn An+l ( s e e f i g u r e 3 ) . I n t h e p r o c e s s , t h e d i s l o c a t i o n i s unpinned a t t h e a t t r a c t i v e o b s t a c l e B pushed

(4)

past the repulsive obstacle R and recaptured by the attractive obstacle C The

n' n'

corresponding potential energy diagram (figure

4)

has four minima that must be taken

into account. At the lowest E , the dislocations interact with the attractive ob-

stacles only. This gives rise to a Debye peak which is E-inde~endent'~). At higher

E , the dislocations are forced to interact with the repulsive obstacles also. Ac-

cording to Schlipf and ~chindlrnayr'~), the dislocation velocity

(9)

is given by

9

= Zw sinh (gb 3 ~ e f / c ~ T )

[l1

where w = jump frequency at a = 0, g 1 (geometrical factor), b = Burgers vector,

c = concentration of obstacles, k = Boltzmann's constant and aeff = effective stress.

If internal stresses are neglected, aeff is composed of the external stress (U) and

the back stress, ( ~ % / b ) d ~ ~ / d x ~ , due to the line tension (Ta). T is taken to be

II

0.5 ~b~ where G is the shear modulus. For E > c ~ T / R E ~ ~ , eqn. [l] defines an E-dependent dislocation mobility and a solution in closed-form is not possible. Assuming interstitial 0 obstacles, c 2 X 1 0 - ~ , and since Young's modulus (E) = 99 GPa, and

the average orientation factor R 1 2/(3n) (for a random assemblage of grain orienta-

tion~), the above inequality predicts €-dependence for E > 4 X l ~ at - 475 K. ~ This is in keeping with the results in figures 1 and 2, indicating that a solution of eqn. [l] is necessary.

POTENTIAL ENERGY E(sl

l

Fig. 3 : Upper diagram shows the basic Fig. 4 : The potential energy of the

jump of the dislocation from one attrac- zig-zag element is plotted as a function

tive obstacle B,, over a repulsive ob- of the reaction path parameter, S.

stacle R,, to an attractive obstacle C,. The lower diagram also defines zig-zag bow-out of a network length (L).

If it is assumed that the average bow-out of the network length (under stress) is given by a cosine function (an excellent approximation when non-linear effects are

negligible(7)), an approximate solution can be obtained for a linearized cyclic

stress, which in turn leads to a closed-form expression for A. The equations given

by Schlipf and Schindlmayr(8) are incorrect. We have repeated the calculations and find

A = (2/n) ( A L ~ ) ( E / G ~ ~ ) [2 in{ (p2+l)'/p}

(5)

C5-322 JOURNAL DE PHYSIQUE

X

where Li2 = -o/ [(ln(l-E))/S]dg (the dilogarithm function), w = 2nf, a = oob3/ckT (uo is the stress amplitude), p = (n/2)/aowr, T = ~ ~ c k ~ / n ~ = ~ w'exp(-U w ~ ~ ~ b ~ ,

eff'

kT) where w' is an attack frequency and Ueff is the effective activation enthalpy,

and B = (p2!l)'/pr. The equations for q1 and q2 are

41 = [(P2+1)'-1~/[(p2+1)'+ll~; 9 2 = S 131 with

s

= [p/{ (p2+1)'-1}l. [(p2cosh2(~B/2w)+l)'-p cosh(n~/2w)].

A plot of AN (normalized A) vs. wr for various values of a. is shown in figure 5.

This family of curves can be described as a peak of almost constant height, which

broadens on the high wr (low T) side as a (or E) increases, while the low UT (high

T) asymptote is unchanged. These curves are qualitatively very similar to the experimental results shown in figure 1.

Fig. 5 : Normalized internal friction as a function of wr for various

values of ao.

To make a more detailed comparison between theory and experiment, we have calculated A vs. T (E const.) and AN vs. E (T const.) curves for the same ranges of E

N

and T explored experimentally. These calculations have been carried out using

eqn. [ 2 ] adapted to the strain distribution in our flexure specimens(9). The para-

meters used in the calculations were: w' = 1 X 1012 S-l, Ueff = 1 eV (simply esti- mated from the peak position), w = 8n, ~b~ = 7.47 eV, c = 2 x 1 0 and L ~ ~ = 1000 b. The activation volume (V) of the process (b times the diamond area swept-out by the

dislocation, figure 3) is 2bZQ, or more simply, V = b3/c = 5000 b3. The calculated

results corresponding to figures 1 and 2 are given in figures 6 and

7,

respectively.

In figure 7, the arrows indicate the approximate range of E explored experimentally

in figure 2. Comparison of the experimental and theoretical curves shows that qualitatively the agreement is very good.

5. Discussion.- The magnitudes of Ueff = 1 eV and V = 5000 b3 are consistent with

(6)

. X..,

g

0.2- a

z 0.1 -

,

' G ' :

i

!

6 7 8

Fig. 6 : Normalized internal

friction as a function of temperature calculated for the various indicated strain ampli- tudes.

Fig. 7 : Normalized internal

friction as a function of strain amplitude calculated for the various temperatures indicated. Arrows indicate the experimental strain amplitude range.

STRAIN AWUTUDE x 106

SSH in a-~r'l~). Fernandez and ~ovolo(~) have attributed an E-dependent A peak at

about 475 K in large-grained Zr to the interactions between oxygen and extended

dislocations. However, their results obtained at 50 kHz yield a much broader peak

(extending to 800 K on the high-T side) than the results in figure 1. Comparison of

their results with theoretical predictions for the unpinning of extended dislocations(13) yields a reasonable value for the binding energy of 1.3 eV, but an unrea-

sonably large dislocation loop length (= 5 X 104 b). In the model presented here,

Ueff is not simply related to the binding energy (see figure 4) but is not incon-

sistent with the high values reported for the binding energy (3y5'14). It should be

noted that the parameters used in the calculations for figures 6 and 7 are not the

only set which give qualitative agreement between theory and experiment. Neverthe- less, the very good agreement obtained using plausible parameters is very promising. It suggests that a quantitative comparison between theory and experiment can be attempted using single crystals (i.e. with R known), if an independent determination

of the network length, L, can be obtained and the concentration of interstitials is

known. REFERENCES

1. R.B. Schwarz and R. Labusch, 3 . Appl. Phys. s(10) (1978) 5174. 2. R.B. Schwarz, Proc. ICSMA 5, Aachen (1979)

2

953.

(7)

CS-324 JOURNAL DE PHYSIQUE

A. Atrens, Scripta Metall.

8

(1974) 401.

L. Fernandez and F. Povolo, J. Nucl. Mater.

66

(1977) 79. R. Schindlmayr and J. Schlipf, Phil. Mag.

3

(1975) 13.

J. Schlipf and R. Schindlmayr, Phil. Mag.

3

(1975) 25.

J. Schlipf and R. Schindlmayr, Proc. ICIFUAS 5, Aachen 1973, Springer-Verlag,

Berlin (1975)

2

439.

K.W. Sprungmann and I.G. Ritchie, AECL-6438, 1980.

L.J. Teutonico, A.V. Granato and K. Liicke, J. Appl. Phys. x(1) (1964) 220. D.G. Blair, T.S. Hutchison and D.H. Rogers, Can. J. Phys.

2

(1971) 633.

P. Lukac, J. Scient. Ind. Res.

2

(1973) 569.

D.G. Blair, T.S. Hutchison and D.H. Rogers, Can. J. Phys.

48

(1970) 2955.