PHENOMENOLOGICAL FEATURES OF THE AMPLITUDE DEPENDENT DISLOCATION DAMPING IN METALS

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PHENOMENOLOGICAL FEATURES OF THE AMPLITUDE DEPENDENT DISLOCATION

DAMPING IN METALS

D. Gelli

To cite this version:

D. Gelli. PHENOMENOLOGICAL FEATURES OF THE AMPLITUDE DEPENDENT DISLOCA- TION DAMPING IN METALS. Journal de Physique Colloques, 1971, 32 (C2), pp.C2-137-C2-143.

�10.1051/jphyscol:1971228�. �jpa-00214554�

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JOURNAL DE PHYSIQUE

Colloque C2, supplkment au no 7 , tome 32, Juillet 1971, page C2-137

PHENOMENOLOGICAL FEATURES

OF THE AMPLITUDE DEPENDENT DISLOCATION DAMPING IN METALS

D. GELLI

Istituto Sperimentale Metalli Leggeri C . P. 129, Novara

Rburnk. -

Des singularitks ou des discontinuitks ont ktk observkes dans un grand nombre de courbes qui expriment le frottement intkrieur en fonction de l'amplitude de dkformation.

Ces courbes ont ktk obtenues avec diffkrents appareillages comprenant deux types de pendules de torsion a basse frkquence et un appareil type Bordoni extensionnel, ayant un intervalle de frkquence 10 4-

50

kilocycles.

Dans le cas le plus gknkral, ces points de singularite divisent les courbes en trois parties caractk- riskes respectivement par une allure linkaire, une allure parabolique et, enfin, une allure exponen- tielle de type Granato-Liicke.

En ce qui concerne les effets des impuretks et de l'ktat technologique de Skchantillon, il a Btk possible de conclure sur la validitk gknkrale de la situation effective suivante

:

-

Dans le cas oh on peut s'attendre a ce que la densitk des impuretks autour des dislocations soit trbs faible (matkriaux trbs purs ou trbs kcrouis) le deuxibme des segments susdit, c'est-A-dire le segment parabolique, est absent, tandis que le domaine d'existence du premier segment (linkaire) tend B faiblir, ou a disparaitre tout a fait. Dans ce cas, on peut dire que la loi de

G.

L. est valable pratiquement dans tout le domaine des amplitudes considkrkes.

-

Lorsque la concentration des atmosphbres des dislocations a de bonnes probabilitks d'&tre caractkriske par des valeurs un peu plus 6levCes (situation d'kcrouissage et de puretk moyenne), on trouve toujours que les segments caractkristiques de la courbe du frottement intkrieur en fonction de Samplitude de dkformation sont prksents tous les trois.

-

Lorsque, enfin, on s'attend a ce que les atmosphbres des dislocations soient considkrablement concentrees (matkriaux plut6t impurs et bien recuits), on trouve que seuls les deux premiers seg- ments de la courbe sont prksents. Dans ce cas il n'y a pas de rectilinkaritk des diagrammes de G. L. (tout au moins, dans le domaine d'amplitude des appareils utilisks).

L'auteur pense que ce comportement est en accord avec un modble de frottement intkrieur par dislocations knonck auparavant [l,

3, 51.

Abstract. -

Some singularities or even discontinuities have been observed in a large number of curves expressing the internal friction versus the amplitude of the alternating deformation.

These curves had been obtained by different apparatuses including two low frequency torsional pendulus and a kilocycle-range, extensional-mode, Bordoni type apparatus.

In the most general case the singularity points divide the curves in three sections characterized by a linear trend, a power-law trend and an exponential Granato-Liicke type trend, respectively.

Concerning the effects of impurities and the technological story of the specimen, the following general behaviour has been ascertained.

-

When the dislocation atmospheres are expected to be most dilute (purest or most hevealy cold worked materials), then the second (parabolic) section of the curves tends to be strongly reduced or even to vanish

;

the range of existence of the initial rectilinear part is strongly reduced in addition and sometimes vanishes too. In these cases the

G.

L. law practically describes the amplitude dependence.

-

When the dislocation atmospheres are expected to be somewhat more concentrated (inter- mediate amounts of cold working or impurity content) one always finds that all the three charac- teristic sections of the amplitude dependence curves are present.

-

When the dislocation atmospheres are expected to be the most dense (carefully annealed, rather impure materials) one finds that only the two first sections (the linear section and the para- bolic or power-law section) are detectable inside the amplitude range of the employed apparatuses.

In this case no more rectilinear Granato-Liicke plot is available.

This behaviour seems to the author to be in a very good agreement with a previously advanced dislocation damping model

[l,

3,

51.

In the present communication the author has in put in evidence in some previous investigations carried mind to outline a regularly appearing set of singula- out a t I. S. M. L. Labs. in Novara by kilocycle internal rities or even discontinuities in the curves expressing friction measurements on aluminium, slightly pure the dependence of the internal friction on the ampli- [l, 21 or doped with small amounts of Cu atoms

tude of vibration.

[3,

4, 51.

Such singularities or discontinuities had been firstly The phenomenological situation which had been

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971228

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C2-138 D. GELLI

is effective in expressing the third segment of the amplitude dependence curve. This is just the law which describes the amplitude dependent part of the damping in the dislocations-pinning points configuration dealt with, in the G. L. model.

The systematic employment of the above plottings has been found to be a useful method for the pheno- menological analysis (i. e. determination of the above parameters) and for the consequent phenomenological description of the amplitude dependence of the internal friction.

envisaged is the one reasumed in figure 1, in which amplitude dependent internal friction data are reported

-

versus strain amplitude

(").

- In the upper part of this figure the typical curve

recl~ltnear when

expressing the dependence of the total logarithmic

6 - 0 y . b

- decrement 6 versus the amplitude y of the alternating

strain is represented. The material is an AlCr solid solution and the plot is linear to linear.

Three singularity points (or discontinuities) can be

(a)

envisaged at 6

=

6,, 6

=

and 6

=

6,, in corres- pondence to three critical vibration amplitudes

yo,

71,

\

The aim of the present communication is to under-

I I

0,1 0,2 0,3 0:4

10-l/y line this fact and to expose the conclusions of an investigation recently carried out on various materials,

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FIG. 1. - Dependence of the internal friction on the maximum (*) Figure 1 could consist simply of a schematic drawing ; amplitude y of the alternating strain, -an example. The basic anyway it seems better to report true experimental results with plottings which are the tolls of the outlined phenomenological the purpose of giving an idea of the fitting of experimental analysis, are reported. The singularities are well advisable and points with the schematic (or ideal) trend. The letter y will the related critical values of the variables are indicated. The be employed (to indicate the amplitude of the strain) whenever parameters a, cc and A are evidently the slopes ^D,while b, dealing with shear strain (torsion pendulus) while letter E will be /3 and B are the ⁽⁽intercepts ^D. employed, when dealing with normal strain (extensional mode).

and y2 (see quoted figure).

Three critical plottings had been deviced to analize the shapes of the curve in each one of the three seg- ments in which the curve itself is subdivided by the above quoted singular points.

The first plotting is just the linear to linear plot under consideration. A rectilinearly increasing segment is visible in its low amplitudes region. This obviously means that the initial increasing part of the curve can be expressed by the law 6

=

6, + (ay + b) where a and b are the slope and the intercept of the straight line, corresponding to this rectilinear part.

The second plotting, is the log-log plotting of the difference 6 - 6,, where 6, is the highest value of the rectilinear segment of the above plot. This new plot contains in turn a rectilinear part ; this means that a power (or parabolic) law of the type 6 - 6 ,

=

Bya is able to describe this segment ; being a and p constant quantitaties.

The third plotting is the classic Granato-Liicke plott-

7

v3 K J

I O - ~ I O - ~ _y ^ing ^[6, 71 applied to the difference 6 - 6, where 6, is

(b)

the quantity indicated in figure 1, on the top of the power law segment. When this G. L. plot is a straight line, one can deduce that the law

6 - 62

=

(B/Y) .exp(-- A/Y)

I 1 1 1 1 I 1 1 1 1 1 1 1 I

, ^,

- -

I O - ~

-

- -

1 -

- -

-

- -

I O - ~ rectilinear when -

-

1

-

^-

-

1 1 1 1 I I 1 1 1 1 1 1 1 I I

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PHENOMENOLOGICAL FEATURES OF THE AMPLITUDE DEPENDENT DISLOCATION C2-139

with the purpose to give an answer to each one of the metals as Ag, Cu and Pt, in the cold worked and two following questions

:

annealed state, have been examined in addition.

1) is the presence of the above singularities and of One part the new lot of measurements had been the related trends of the curve a general fact, obtained by the previous kilocycle range-extensional mode Bordoni type-apparatus (see quoted papers 2) in which way the phenomenological picture is 1 + 5) while another part had been obtained by two affected by the chemical purity and the technological different kinds of (low frequency) torsion pendulus.

story (or state) of the specimen

?

Of course the introduction of the last additional For this purpose, about 50 amplitude dependent

internal friction curves pertaining to different binary solid solutions (including AI-Cu, -Mg, -Zn, -Si, -Mn, -CO, -Cr, -Zr) and to different technological states, have been considered. Slightly pure further base-

measuring techniques adds further degrees of freedom concerning mode and frequency of vibration and range of strain amplitude and thus it renders more general the validity of the conclusions which can be drawn from the investigation.

Well, after having examined such an ensemble of results one can surely conclude that the answer to question 1 about the general existence of singularities and variously analytically describable segments, gets an altoghether positive answer.

At this point it should be advisable to report the documentation supporting this statement but it is impossible to present here all the examined data

;

furtherly it seems to be preferable to postpone the presentation of the selected examples of the outlined behaviour to the next discussion of question n. 2 (about the effects of the chemistry and story of the specimen).

What seems instead useful, presently, is to underline some differences which have been observed among the discontinuities outlined above and the possibility to classify them in

:

-

smooth discontinuities, - sharp discontinuities, - step discontinuities.

Some examples there of are given in the following figures. Figure

2a, c,

are examples of smooth, sharp

FIG. 2. - Examples of singularities at medium amplitude and decrement levels (near 81 or y l and 61) : a) smooth discontinuity ; 6 ) sharp discontinuity ; c) step discontinuity.

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C2- 140 D. GELLI

and step discontinuities at low amplitude and damping level (near or

y1

and 6,).

Figure 3a, c, are examples of analogous disconti- nuities at higher amplitudes and damping levels (near Y z and

^~2).

The sharp discontinuities are far the most common

;

the smooth and step ones are iess common (about tenfifteen percent altoghether).

Going now to consider the second question (about purity and state of the specimen), it is to be noted that the influence of purity is essentially a matter of how many pinning points (especially foreign atoms) are at disposal of each dislocation. Thus the ratio

between the concentration of pinning points and the concentration of dislocations is of importance.

Furthermore it is important the question whether the foreign atoms (or, generally, the pinning agents) have been able or not to migrate close to the disloca- tions to exercise their pinning effect.

In other words it is more effective to speak in terms of clearness of the neighbourhood of the dislocations rather than simply in terms of the average chemical purity. (This is true every time one is dealing, not only with the dislocation damping but also with other dislocations related phenomena, like solution harden- ing or some other hardening processes).

This dislocation clearness depends on the story of the specimen : for instance, a given specimen in the cold worked state should generally be expected to present a dislocation clearness much higher than the same specimen in the well annealed state. This is due to the decrease of the ratio

:

dislocation density

3z =

pinning points concentration as a consequence of the annealing treatment.

Now the rather high number of compositions and states which have been examined, allowed one to consider just a reasonably wide latitude of situations regarding purity ratios and the topological distribution of the pinning agents with regard to the dislocations.

The conclusions drawn from this inquiery lead the following statement to be practically ascertained.

1. When the dislocations neighbourhoods are likely to be mostly clear, as it happens with the purest materials or with the most heavily cold worked mate

F IG. 3. - Examples of singularities at higher amplitude and decrement levels (near ez or y z and 67.) : a) smooth discontinuity ; b) sharp discontinuity ; c) step discontinuity. (The log-log plots have been here employed as the presence of the rectilinear segment renders the discontinuities more evident).

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PHENOMENOLOGICAL FEATURES OF THE AMPLITUDE DEPENDENT DISLOCATION C2-141

rials, i. e. when the ratio n between the overall impu-

rity concentration c and the dislocation concentration F, is expected to be the lowest, or the pinning points are expected not having been able to migrate to the dislocations, the previously quoted (third) exponential segment, of the curve extends practically to the whole curve, i. e. the Granato and Liicke law fully describes the amplitude dependence. The second (parabolic) section of the curve vanishes and also the range of existence of the initial rectilinear part is strongly reduced or, sometimes, vanishes too.

Examples therefore are given in figure 4a and 5a.

In figure 4a, which pertains to well annealed, zone refined aluminium containing 3 p. p. m. Si, the situa- tion outlined above (absence of the parabolic segment) is determined by the high chemical purity of the

specimen. The rectilinear and the G. L. segments are, on the other side, well detectable. In figure 4b (AlCo 0,47 % cold drawn 98 %) it is represented the case of a high dislocation clearness due, not especially to the chemical purity of the specimen but rather to the high dislocation density (which is maintained also due to dispersed CO-bearing compounds). Also in the present case the parabolic section of the curve has vanished.

The rigorous rectilinearity of the G . L. plots, reported in the above figure 4a and 5a, attests that the trend is

:

as outlined.

2. When the concentration of the dislocation atmos- phere is likely to be an intermediate one e. g. (i) in a slightly pure material or (ii) in a very moderately deformed material or (ii) in a more heavily deformed material which has undergone an advanced recovery

10-4

or a partial annealing or a quench from a not too high temperature, then the three segments of the amplitude dependance curve are all present.

Figure 4b shows the amplitude dependence curve

R1

pertaining to a specimen similar to the preceeding one, from zone refined aluminium, with the difference that the impurity content is now 30 p. p. m. Si (instead of

2 ^-

being 3 p. p. m.). In the present case the parabolic seg-

- 10-5

ment has reappeared

;

all the three segments can be seen.

- -

In figure 5b the effect of the thickening of the dislo-

-

cation atmospheres, due to a partial annealing, is shown. In this figure the specimen A1 0,48 % CO, previously presented in the cold worked state is now shown after an annealing treatment at 250 OC. The parabolic section reappeared and the three segments

0.8,

I

, , ^{I O - ~}

^{l l} ^l ^l ^l ^l ^l ^l ^l ^l

^{I O - ~}

^l ^, ^8.3xE^l ^l

^10-L

FIG. 4. - Effect of the chemical purity on the (( composition )) of the amplitude-dependence curve in Si-doped, zone refined, well annealed aluminium. a) Very pure specimen ; only the rectilinear segment and the G. L. segment are detectable.

b) Intermediate chemical purity : all the provided segments (rectilinear, parabolic and G . L. segments) are present. c) Lowest- purity specimen : the rectilinear and parabolic contributions invade the full measurable range ; no more G. L. segment is available.

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C2-142 D. GELLI

are again all present together. The correct "trends of range of the apparatuses. On the other side, no recti- the second and third segments of the curves of figure linear Granato-Liicke plot is normally available, i. e., 7a and 7b is attested by the log-log plots of 3 - 6 , and the third part of the amplitude dependence curve tends

the G . L. plots of 6 - 6,. to disappear.

3. Finally, when the dislocation atmospheres are An example, of this situation, with regard to the expected to be most dense as a consequence of the effect of the composition, is given by figure 4c, in composition and/or of the applied treatments, e, g. in which the results obtained on a specimen analogous impure orland carefully annealed materials, one finds to some preceeding ones, but containing 1 000 p. p. m.

that only the two first sections (the linear section and Si (instead of 3 or 30) are represented. In this case it the parabolic or power-law section) of the dislocation is well clear that the G. L. segment of the curve has damping curves, are detectable inside the amplitude disappeared (or eventually has been displaced to much

FIG. 5 . - Effects of annealing on an initially cold drawn specimen (from commercial superpure aluminium) : a) in the cold worked specimen (low clouds density) only the rectilinear and the G. L'segments are available ; b) after annealing from 250 O C ,

the parabolic low appears ; c) in the well annealed state (600 OC) the G. L. segment disappear being replaced by an extended power law segment, as it happens in the

I s

pure specimen of preceeding figure 4.

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PHENOMENOLOGICAL FEATURES OF THE AMPLITUDE DEPENDENT

DISLOCATION

C2- 143

higher strain amplitudes, beyond the range of the

apparatus).

An analogous effect can be observed as a conse- quence of further increased cloud density (and decre- ased dislocation density) due to annealing. The cold drawn specimen A1 0.48 CO, previously considered after annealing from 250

OC,

is shown in figure 5c, after an annealing treatment from 600

OC.

It is well advisable that the G. L. segment disappeared and its amplitude dependence curve is composed only by the rectilinear and the parabolic segments.

The present communication is related only to the phenomenological analysis of the effects and to the employment of the amplitude dependence internal

friction curves as a tool to study some metallurgical or solid state processes.

N o time has been allowed for the physical interpre- tation of the phenomena which anyway are all in a very good qualitative agreement with a previously advanced model

(1).

The quantitative implications will be discussed sepa- ratelly.

AcknowIedgements. - The author is grateful to Professor F. Gatto Director of I. S . M. L. for criticism and for the permission t o publish the results.

Thanks are also due to Messers L. Reccher and E. d'Intino for their technical cooperation.

References [l]

GELLI (D.), A Qualitative Model for Amplitude

Dependent Dislocation Damping

^D.J . Appl. Phy- sics, 1962, 33, 1547.

[2]

PANSERI

(C.),

FEDERIGHI (T.), GELLI (D.), ((Effect of Quenching on Room-Temperature Dislocation Damping in Aluminium

^N. Suppl. a1 Nuovo Cimento, Skrie X, 1962, 26, 23.

[3]

GELLI (D.), ((Dislocation Damping of Zone Refined Aluminium Doped with Cu

^D.Phys. Stat. Sol., 1965, 12, 829.

[4]

GELLI (D.),

((

Some Effects of Quenching on the Dislo- cation Damping of Zone Refined Aluminium Containing Cu Atoms as Pinning Points

^)).Phys.

Stat. Sol., 1965, 12,

K

109

(Short Notes).

151