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ON THE BACKGROUND INTERNAL FRICTION IN ALLOYS
P. Feltham
To cite this version:
P. Feltham. ON THE BACKGROUND INTERNAL FRICTION IN ALLOYS. Journal de Physique
Colloques, 1971, 32 (C2), pp.C2-187-C2-189. �10.1051/jphyscol:1971242�. �jpa-00214568�
JOURNAL DE PHYSIQUE Colloque C2, supplbment au no 7 , tome 32, Juillet 1971, page C2- 187
ON THE BACKGROUND INTERNAL FRICTION IN ALLOYS
P. FELTHAM
Brunel University, London, W. 3
Rbum6. - Cette etude montre que les oscillations des petits arcs de dislocations, effectuQs par la diffusion des decrochements geomktriques, peuvent bien expliquer l'apparition du frotte- ment intkrieur (( du fond 1) dans les solutions solides. L'intensitd du frottement, et sa variation avec tempkature, dkformation prkalable, concentration de l'alliage et amplitude, calculQs ?I l'aide de ce modde, se trouvent en assez bon accord avec l'experience.
Abstract. - The oscillation of small dislocation segments, facilitated by the diffusion of geo- metrical kinks, is shown to account satisfactorily for the (( background u internal friction in solid solutions. The magnitude of the internal friction, and its dependence on temperature, pre-strain, alloy content and amplitude, evaluated by use of the model, are in good agreement with observa- tions.
Contrasted with the numerous studies of internal friction peaks in crystalline materials, the background internal friction has received little attention. Routbort and Sack [1], who studied it in single crystals and polycrystals of aluminium, copper and magnesium at 1 Hz to 40 kHz as function of plastic deformation and gamma-ray as well as neutron irradiation, found it to be amplitude independent and, except for a slight decrease at the highest frequencies in the range, practically frequency independent. They remark that this contribution to the internal friction cannot be explained in terms of the usual (( pinning )) models of the Kohler-Granato-Liicke type.
Heiple and Birnbaum [2], who made measurements on high-purity copper single crystals in the kHz and MHz ranges, came to similar conclusions concerning the kHz oscillations ; at MHz frequencies they were able to interpret the results by a (( vibrating string )>
model, which they used to explain a peak observed at about 1 . 7 MHz.
Spears [3] recently investigated the background inter- nal friction in a-brasses containing 10-30 atomic percent of zinc at 120-300 OK in the kHz range, also using deformation as a variable. Theoretical considerations suggest [4, 51 that breakaway of dislocations, such as is generally visualised to occur in dilute alloys at sufficiently high amplitudes, is unlikely to occur at strain amplitudes of the order of used in these experiments. In fact, Spears confirmed that the inter- nal friction was amplitude independent, consistent with the absence of classical breakaway processes.
It was also rather insensitive to frequency changes, thus resembling pure metals in this respect as well.
Other observations made by Spears, which any model of the process would have to account for, are the gradual, near-linear, decrease of the internal friction
(Q-l) with decreasing temperature, with the Q - I
versus T relation extrapolating to the origin, then its enhancement by prestrain in the linear hardening range, and its subsequent reduction by further defor- mation. Finally, it should explain the decrease of Q - I with increasing alloy content.
All the results referred to indicate that limited dis- location movement must be possible without (( break- away )) at the low amplitudes used, and that it is not adequately described by current theories of dislo- cation-induced internal friction. It seems however that the stress-aided diffusion of geometrical kinks should facilitate the required type of displacements.
Below we outline a model based on such a mechanism.
1. Theory.
-
Referring to figure 1, we shall assume that dislocation segments pinned at their ends by jogs, nodes, high local internal stresses and other structural heterogeneities, can migrate a limited amount under the influence of local stress perturbations due to the applied harmonic force, as has been proposed for example in a discussion of the Bordoni peak in pure metals by the present author [6].Displacements of this type, i. e. from left to right in figure 1, can occur by the diffusion of kinks without necessitating an increase in the length of the segment pinned at A and B. In the brasses considered by Spears, which we shall refer to, zinc atoms will be spaced on the average 2-4 b apart on any dislocation, and they will exercise a drag on it akin to an enhanced lattice friction. A kink would therefore diffuse under some standard conditions less easily in the brasses than in copper.
Now considering a (( representative >> segment with an orientation in the glide plane in which I represents an averaged mean spacing between geometrical kinks
13
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971242
P. FELTHAM
in the absence of an applied stress, we shall express the instantaneous mean kink velocity by Einstein's relation
where x is the mean displacement from the equili- brium positions which they occupy at the instant at which the stress is applied. As shown in figure 1, x is measured along the projection of the segment onto the direction of the Peierls trough close to which the segment AB lies. The average force on a kink will be taken to be zb2, where b is the lattice spacing and z the shear stress acting on it in the slip plane. As a result of the bowing of the segment, as it tends to the limi- ting position ACB, the most probable spacing between kinks will diminish approximately in the ratio 1/(1
-
x).As a result of the Coulombic repulsion between kinks of equal signs [8], their drift velocity will also decrease as the segment becomes curved due to the movement to the right. For our purposes it will be adequate to represent this effect through an incre- ase in the energy barrier to their drift motion, and we shall write the aiffusion coefficient in the form D(x) = vb2 exp
[ - - ( --- l X (2)
where v is of order 10'' Hz. Further, we shall confine
our attention to sufficiently small displacements to
enable us to replace the fraction 1/(1 -
x) by (1 +
x/E).
One then obtains as the solution of eq. (1) :
where z and T have been assumed constant.
Now, on taking z = G, Gb3 = 5 eV,
T = 150 OK, o = 12.5 kHz, also using the estimates I = lo3 b and u, = lo-' eV, one finds that for t = 7110, i. e. corres- ponding to a half cycle of a << square >> wave, the term in rectangular brackets in eq. (3) is equal to about 1.15. We shall therefore replace it here by the first term of its series expansion, so that then
The average number of kinks participating in the diffusion process is p/A, where p is the ((effective D dislocation density i. e. comprising only dislocations containing kinks, i. e. not lying in Peierls valleys [9].
The area swept out by the kinks in the displacement x is (p/R.) bx, and the anelastic strain resulting from these displacements is (p/;l) b2 X. As this is appreciably less than the elastic strain z/G, the modulus defect for t = 71/0 may be taken equal to the ratio of the ane- lastic to the elastic strain, so that, using eq. (4), one obtains
Under the experimental conditions used, where seve- ral slip systems are operative, the effective modulus defect is equal to A , O, where 0 is the well known orientation factor 110, 111, which we shall set equal to 0.1. Then, with the above values of the parameters, also taking p = 1 0 % m - ~ for a deformation of a few percent, and putting R. = 10 b, b = 2.5 x l o e 8 cm, one obtains for the effective modulus defect from eq. (5) a value of 0.01 5.
In order to determine the internal friction, Q-I, one has to consider that the segment AB as a whole will respond to the periodic applied stress as if it were an elastic string with a <( line tension )) about an order of magnitude less than that customarily asstimed for dislocations in continuum treatments, i. e. where kinks are not allowed for [8, 91. The loss maximum would therefore be expected to occur at a frequency o, in the MHz range, which is consistent with the results on copper [2] referred to above. Thus, on writing
not allowing for the fact that a distribution rather than a single value of o, would be more appropriate, and neglecting ( ~ / o , ) ~ compared with 1, one obtains
As A, is inveresely proportional to o , the internal friction given by eq. (7) is frequency independent within the limits of the variables set by the approximations made in deriving that result. Now, taking w/o, = 100, with o = 12.5 kHz, as used by Spears [3], Q - I is found to be equal to approximately 1.5 x which agrees well with the experimental results.
2. Discussion and conclusions. - Although, in view of the simplicity of the model and the empiricism involved in deriving eq. (7), the good numerical agree- ment with observations is certainly fortuitous, the functional forms of eq. (6) and (7) account for the experimentally established trends. Thus, they require that :
a) The internal friction should be amplitude inde- pendent,
ON THE BACKGROUND INTERNAL FRICTION IN ALLOYS C2-189 b) and its dependence on frequency should be
small.
c ) It should tend to zero as T + 0, a trend consis- tent with the extrapolations to low temperatures shown by Spears.
d) Alloying should lead to an increased drag on the dislocations and, hence, to an increase in u,. The impli- cations of this agree with the observed effect of increa- sed zinc content on Q-I in brasses [3].
e) The almost linear initial increase of Q-I with prestrain [3] is explicable by an increase in the effective
dislocation density p, referred to in eq. (5). Network formation, jogs, point defects, high internal stress fields, etc. would tend to reduce p at large strains, i. e.
in the third stage of work-hardening. This provides a qualitative explanation of the decrease of Q - I at large deformations.
A weakness of the present treatment is the empi- rical transition from eq. (5) to eq. (6) A more realistic model would have to take account of the spectrum of energy barriers due to the heterogeneous internal stress field.
References
[l] ROUTBORT (J. L.) and SACK (H. S.), J . Appl. Phys., [61 FELTHAM (P.), Phil. Mag., 1966, 13, 913.
1966, 37, 4803. [7] BRAILSFORD (A.), Phys. Rev., 1965, 139A, 1813.
[2] HEIPLE (C. R.) and BIRNBAUM (H. K.), J. Appl. Phys., [8] SEEGER (A.) and SCHILLER (P.), Physical Acoustics,
1967, 38, 3294. 1966, 3A, 361.
[3] SPEARS (J. S.), Lausanne Conference on Anelastic [91 THOMPSON (D. 0.) and HOLMES (D. K.), J . Appl.
Effects due to Defects and Phase Transformations Phys., 1959, 30, 525.
in Solids, 1970 (This issue). [lo] ALERS (G. A.) and THOMPSON (D. O.), J. Appl. Phys., [4] FELTHAM (P.), Brit. J. Appl. Phys., 1968, 1, 303. 1961, 32,283.
151 LEHMANN (G.), Brit. J. Appl. Phys., 1969, 2, 126. [ll] YAKOVLEV (L. A.), SOV. Phys. Acoustics, 1965, 11, 197
