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Submitted on 1 Jan 1981
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INTERNAL FRICTION IN CONCENTRATED
ALLOYS
C. Spears
To cite this version:
CoZZoque suppZdment au nO1O, Tome 4 2 , octobre 1981
INTERNAL FRICTION IN CONCENTRATED ALLOYS
C.J. SpearsMiddZesex Polytechnic, MiddZesex, England
Abstract.- Internal friction in concentrated alloys at k H z frequencies and at temperatures from 120
-
3 0 0 K has beenstudied in alpha-and beta-brasses as a function of grain size, temperature and pre-strain, and a statistical model has been proposed to account for the observations.
1. Introduction.- I n a paper given at the Aachen Conference (1) a tentative model was proposed to account for the strong concentration dependence and for ordering effects at compositions close to CusZn observed in polycrystalline a,-brasses containing 0
-
3 0 atomic $ of zinc. Further experimental results, again taken at strain amplitudes of about 1 o - ~ , at 12.5 k H z , and a new model of the internal friction are considered in the present paper. The theoretical interpretation of the internal friction used in (1) was based on a n assumed broad spectrum of relaxation times, associated with the thermally activated movement of groups of geometric kinks on dislocation segments of various lengths (2). Although a contribution from such oscillations is not to be excluded, the rather-large magnitude of the background internal friction (&-l a I O - ~-
I O - ~ ) and its frequency independence over many decades of frequency, i.e. from a few Hz to well into the kHz range(3)
are difficult to reconcile with the small internal friction implied by loss from a narrow band of the relaxation spectrum around the resonance frequency of the specimen.We
(4)
shall show that the observations are quite well accounted for in terms of a model i n which the internal friction is hysteretic, arising from the non-resonant bowing of dislocation segments. The effect of alloying will be considered to be the reduction in the number of segments able to bow-out under the applied stress.2. Experimental.- The experiments which were a n extension of earlier work
(1)(5)
on solute compositions up to 30 atomic%
of zinc were again carried out on polycrystals containing 35,39,41 and 50 atomic%
of zinc i.e. concentrations within the range of D , w/P and 8-brasses.C5-278 JOURNAL DE PHYSIQUE
The main impurities were as beforz, viz. traces of iron, tin and bismuth. Rod shaped specimens were oscillated longitudinally, it was assuqed, 3s in the previous woric, the internal friction Q-'
,
was amplitude independent. Within the tempereture range examined,1 20-330K, the temperature dependence of 2-l was ri~onotone
.
elpha-bress as function of Tensile Strein at 160K and 280K.
2 4 6 E%
The relationship of
2iix,
corresponding to peaks in isothermsat 280K of the type shown in Fig.1, with zinc content is given i n F'ig.2.@
, correct F& dmul of ordung fra.
Q)- A a(f- u p ~ ~ * ) ~ l a t ~-133Kcvd€-4% Q'foruwentthwrg
\-
A .,
Fig.2 : Dependence of Q - ~ on zinc content, empty circles10 2 0 30 4 0 5 0 denote current work.
+
Zn (at)%in the second half-cycle, which m y not necessarily be so in solid solutions then, with n vibrating loops per unit volume, the loss per cycle will be Aru = znZ(aO2La/~). On writing for the density of vibrat-
-
ing disloc2.tions N = n L then, with the usuel orientation f-ctor CV
( F ; : ~ . l ) , one has
c;1
S C U N ~ ~ ( 1the subscript ' 0 ' denoting pure netal.
The loss cpn be seen to be frequency pnd amplitude independent, and with reasonable values of N and
E
it is of the observed mag- nitude ( l ) , it will depend on the deformation of the m?.terial, i.e. through N andc.
4. Internal Friction in Concentrated Alloys.- We now consider a random dispersion of alloy atoms introduced into the metal, resulting in a concentreted solid solution. If, a s we shall assume c > 1 atomic
%,
the average p ~ n n i n g point spacing on dislocations due to solutes, neglecting any Cottrell effects, will be less than b/J0.01, i.e. lob, which will be by ?bout a factor of 100 smaller than referred to above. With the dislocation density N , unchanged by the addition of solute, substitution of lob forL
in eq ( ?1
would lead (with N N 10':
to 2-I of about 1 0-' i.e. several orders smaller than the observed vrlue (Fig.1). If however the hysteretic process represented by eq (1) preveils in the concentrated alloys, one has to assume that, 2s e result of the rendo.mness of the pinning-point distribution i n the crystal, and the spatial heterogeniety of the internal stress, 'weak' points will exist at the dislocstions, so thzt some segments may 'break- out' of the pinning constellatiorz a.nd bow out a s in tbe pure rzretel, the effective pinning-points of such loops corresponding to the length-
L. I n this light the effect of elloying is the immobilisation of some, but not all, loops bordered by the 'strong1 pinners assumed to determine L. Following a suggestion by Feltham ( 4 ) we shall now outline a model for the break-away criterion from the alloy pinners; this must necessarily be rether simple at the present stage of study.
5.
Pinning-point Distribution.- We first consider an exponential alloy-
pinning-point distribution on a length(L
>>a ) ,
such thatn
(L)
d.4 = nL exp ( - ) d L-
-
-
-
there being nL =
L/R
( 3 )
C5-280 JOURNAL DE PHYSIQUE
The expectction v-lue
3
of ! is taken equal to b/ Jc. Breakout from the alloy pinning points will be assumed possible for L segments containing at least one A - spacing equal to or greater than a cert3in critic.z.1 length &cri t
'
formally associated with a 'weak point1 concen- tra tion through Lcri = b/ /ccrit ( 4 :The number of pinning points per
-
segment for which t 2L
crit is f.n_, where
and for >>
3
this reduces to-
fnL =
-
L exp( -
tcrit/T )a
For breakout there must be at least one weak spot on the 1-segment considered,i.e.
f.n = l
L ( 7 ) 9 so that from eq (3),
(6)
and(7)
-
-
f = exp (-acrit/i
) ,
andacrit
= 1 I n(E/?
) ( 8 )If typically
3
= To, i.e. for c = 0.1, and L = 1004 then eq (8) yields f--
10-', ccrit c4%
end lcrit l3b.-
-
-
Lcrit -
a
exp (acrit/e
) =2
expJC
( 9 )JC
G c r i tWe shall here assume that 1 for a random solid solution is crit
known, remembering 3lso that
2
= b/Jc then eq (8) facilitates the-
evaluation of L crit, i.e. segments with L 2 L
crit would bow out, while those with g L would not. I n the above example L = 100 $ =
crit crit
Lcrit
L
giving where X =L/E
-X where4
=3
e (x2+ 2x+
2) X now = L/X
cri t XWe see that for small X, (x2
+
2x+
2) e,
so that @ a 1 . This means that with an alloy only lightly deformed(L >>
L ) almost allcri t
L-segments will contribute to the internal friction;
6
will drop-
r?.pidly only if Lcrit is of the same order of magnitude as L. Bowever, the assumption of r. rc.ndom solid solution is not ~pplicable to dilute alloys (say c < 1 atomic '$) as Cottrell effects
(7)
would enhance the solute concentrrtion above the nominal vzlue, close to the dislocation cores. We c?.n therefore use eq(l1) only for concentreted alloy(possibly c > ? atomic
$).
-
-
In eq (1 2) X = Lcrit/L = exp
(c
) from eq ( 9 )-
L ( 'crid
-
with, say, Lcrit v l3b, L = 1000b and ccrit =
*$
we have X C 1 for c30
atomic'$
so that one can write eq (12) in the simple form?-L Q;' (1
-
(14)
2,;' need not here correspond to the internal friction for copper, as we cannot extrapolate to dilute solutions or pure metzls from the model.
-1
Previous work
(I
) geve a Q, value of6
X Io - ~
for polycrystalline copper.C5-282 JOURNAL DE PHYSIQLJE
discrepancies bet-ween 2-I and the theoretical curve (Fig.2) suggest max
that
a) Cottrell locking may not be negligible even at relatively high con- centrations (i.e. experiment gives lower values than theory). b) comparison of experirnentz.1 date represented by 3 - I with the theo-
max
'
reticel concentration dependence of &-l (eq 12) is tentative, for &-l
depends on the deformation of the material in a complex manner. Fig.1
-
may refer to a different L for every concentrz.tion, and c) the theory, though promising, needs further development.
The reduction of Q-' due to ordering can be understood if it is con- sidered that the more uniform distribution of the zinc ctoms through- out the lattice reduces the Cottrell effect, i.e. the concentrz.tion of zinc atoms near dislocation cores is less in the ordered than in the rsndonl structure
.
7.
References( 1 ) C J Spears. Proc.5t.h 1nternat.Conf.Int.Friction and Ultrasonic Attenuation in Cryst.Solid. Vol.l,p.363, M.Springer, Berlin
