Amplitude independent internal friction and modulus defect in polycrystalline copper at room temperature

Scripta METALLURGICA

Vol. 7,pp. 529-534, 1973 Printed in the United States

AMPLITUDE

II~DEPENDENT

MODULUS DEFECT

Pergamon

INTERNAL FRICTION

IU POLYCRYSTALLINE

Press,

Inc

AND

COPPER

AT ROOM TEMPERATURE 1) DEPENDENCE

K. BelCher, Sektion

UPON PLASTIC

E. Biller,

and D. LGbbers +

Physik der Universit~t

Lehrstuhl M~nchen,

generally LGcke

Germany

properties

attributed

(GL)

According

amplitude

independent

MD, at sufficiently

14, 1973)

in highly purified metals at room temperature

to the movement

(I) has greatly

phenomena.

MGnchen,

Prof. Rollwagen,

(Received March

The anelastic

DEFORMATION

of dislocations.

contributed

to this theory components

The theory of Granato-

to a theoretical

the following

understanding

expressions

of the decrement, ~

are

of these

hold for the

, and the modulus defect,

low frequencies:

MD = E e t - E E

klflAL 2 = k2~ A L4 B

whereby,

Eel is the ideal Young's modulus,

k I and k 2 are constants, length of vibrating

A

is the dislocation

dislocation

segments

frequency, B is the viscous damping is an orientation factor. If a periodic then the resulting

+ present

address:

stress~

E is the measured

~

:

L is the average

(loop length),uO

constant

is the vibrational

for dislocation

movement

and Q

60. e i ~ t, operates within a volume element,

strain is [ =~0"e l( ~ t

Phys. Chem.

density,

Young's modulus,

Institut

-~)=

( ~el + ~disl

der Universit~t

529

- i ~ dis2)

MGnster,

Germany

ei~ t

530

AMPLITUDE

INDEPENDENT

INTERNAL FRICTION

For the m e a s u r e d d e c r e m e n t , ~ : ~ ' t a n ~

IN C O P P E R

, the result

is then

On the other hand, one finds by closer e x a m i n a t i o n decrement

~

, g i v e n by the theory of GL is ~ = ~"

ship then b e t w e e n the two d e c r e m e n t s

is given by

(I)

Vol.

7, No.

5

6 :~

Ed's2 Eel + Ed,s I that the value for the

Edls2 Eel

. The relation-

I + MD

E disl Eel

because MD=

The d e c i s i o n w h e t h e r or not to take into c o n s i d e r a t i o n the difference between

$

and

the modulus

~

depends on the m a g n i t u d e of the modulus defect.

defect,

as here described,

as O.1 and the a c c u r a c y of m e a s u r e m e n t

has been m e a s u r e d

of the decrement was considerably

b e t t e r than 10%, we have always d i s t i n g u i s h ~ b e t w e e n With this in mind, we have always from the m e a s u r e d decrement, 6

Because

up to values as high

calculated

the two decrements.

the value, ~

= 6.(

I + MD),

, and the m e a s u r e d modulus defect,

MD, in order

to achieve a more exact i n t e r p r e t a t i o n of our measurements. We have m e a s u r e d the internal crystalline

f r i c t i o n and the modulus defect

copper in the lower kIIz frequency range.

report the first part of the experiment,

in which the p r o p e r t i e s

m a t e r i a l were changed by means of plastic deformation. tion, we varied a d d i t i o n a l l y paper

the frequency.

in poly-

In this communication,

we

of" the

In a second communica-

This is r e p o r t e d in the f o l l o w i n g

(2).

The m e a s u r e m e n t s were p e r f o r m e d by means of the resonant F 6 r s t e r E l a s t o m a t and an a d d i t i o n a l

bar method with a

capacitive d e t e c t i o n system.

The

c y l i n d r i c a l rods have a length of 200 mm and a d i a m e t e r of 7 mm. E x c i t a t i o n was in the second l o n g i t u d i n a l v i b r a t i o n a l mode of about curve.

20 kHz. The decrement was d e t e r m i n e d

(first overtone)

at a frequency

from the width of the r e s o n a n c e

The c a l c u l a t i o n of the modulus defect of the plastic d e f o r m e d rod was

always based on the Young's modulus a m p l i t u d e was about

E = 1.10 -8 . The amplitude

values in this range was assured. temperature.

The v i b r a t i o n

i n d e p e n d e n c e of the m e a s u r e d

M e a s u r e m e n t s w e r e p e r f o r m e d in air at room

The m a t e r i a l was 99.98 % purity copper s u p p l i e d b y

b r G c k e r Kupfer- und Drahtwerke. at 700 ° C in vacuo

(10 -4 Torr)

the Osna-

The samples were first annealed for 2.5

hours

and then were stepwise p l a s t i c a l l y d e f o r m e d

t e n s i o n up to a m a x i m u m of 4 %.Fig. obtained.

in the u n d e f o r m e d state.

1 shows the s t r e s s - s t r a i n

The average grain size was 0.05 ram. The m e a s u r e m e n t s

in

curve thus of d e c r e m e n t

and modulus were always p e r f o r m e d 24 hours after each d e f o r m a t i o n step,

in order

to a l l o w time dependent effects as much as possible to come to rest. The results shown in the figures were obtained with a single rod. tive m e a s u r e m e n t s with other samples yielded q u a l i t a t i v e l y curves w i t h d e v i a t i o n s in the q u a n t i t a t i v e magnitudes.

Fig.

similarly

Comparashaped

2 shows the

VOI.

7, No.

5

AMPLITUDE

INDEPENDENT

INTERNAL FRICTION

IN COPPER

(i)

l e [mk--~m2] 15-

10"

5I0 ,i

~

lb

:~o FIG.

Stress-strain

~'o lo 3 Ep

~o

1 curve of p o l y c r y s t a l l i n e

copper.

I 1036

I03MD I

~5- r"X,

-15o

x

10-

X

.100 x

~/~

ulus delect x

1 O~

01

g

10

20

30

iO

103 Ep

-10 0

FIG. 2 Internal

friction, 6 , and modulus defect, MD, in

polycrystalline

copper vs plastic

strain,Ep

. The

point at the lower part of the ordinate is the decrement in the u n d e f o r m e d

state.

531

532

AMPLITUDEINDEPENDENT INTERNAL FRICTION IN COPPER (1)

I

(MD)_~2

4

+

15

,-

I

01./

/

Fzs.

3

(MD) 2 / /% vs plastic strain, ~p.

I M_qD A 30.

2o

"

¢

~0.

~

MD_L-Z

FIG. 4 MD / ~ vs plastic strain, ~p

Vol.

7, No.

5

Vol.

7, No.

amplitude

5

AMPLITUDE

independent

INDEPENDENT

components

of the decrement

as a f u n c t i o n of the degree of plastic a sharp maximum,

whereas

begins to d e c r e a s e

IN COPPER

(i)

533

and the modulus defect both

deformation.

The damping passes

through

the m o d u l u s defect increases through a large range and

only at higher degrees of cold work. A similar b e h a v i o u r has

also been r e p o r t e d by other authors In this study,

INTERNAL FRICTION

(3).

the simultaneous m e a s u r e m e n t

of the decrement

as well as the

m o d u l u s defect both as a function of the degree of d e f o r m a t i o n and the a p p l i c a t i o n of the theory of GL to these m e a s u r e m e n t s

enable us to determine

the d e f o r m a t i o n d e p e n d e n c e of the d i s l o c a t i o n density, A therefrom,

of the loop length,

A should be p r o p o r t i o n a l two quotients, fig.

L, because,

to

(MD) 2

, and separately

according to the theory of GL,

and L -2 p r o p o r t i o n a l

to

plotted against the degree of deformation, ~p,

MD

. These

are shown in

3 and 4. Both plots are linear over a wide range. These results appear to be significant.

~p, w h i c h we find,

The p r o p o r t i o n a l i t y

between A

and

is in good a g r e e m e n t with the results obtained by

electronmicroscc,pic m e a s u r e m e n t s

by Bailey

would expect a similar b e h a v i o u r

for the reciprocal

(4) and Essmann et al.

(5). One

square of the loop length,

L -2, if L were equal to the network length, L N , because in general L N - 2 ~ A F r o m the amplitude have observed, present,

we have to conclude

properties,

which we

that additional pinning points must be

so that the loop length cannot be identical with the network

At the most, -2 and L N A

and time d e p e n d e n c e of the anelastic

length.

one can presume that a linear r e l a t i o n s h i p may exist b e t w e e n L -2

d e v i a t i o n from the linear occurs with degrees of d e f o r m a t i o n greater than

2%. This d e v i a t i o n may result

from a faulty d e t e r m i n a t i o n of the ~odulus

defect as a result of a change in texture.

Another possible e x p l a n a t i o n for

this d e v i a t i o n from linearity may be that with a d i s l o c a t i o n density which is to high,

the d i s l o c a t i o n s

deviations

interact with each other, with the result

from the p r e d i c t i o n s

other hand, M c D o n a l d and Fiore aluminum,

that

of the theory of GL can be expected.

(On the

(6), as a result of their experiments with

come to the conclusion that the model of GL holds true only at high

degrees of d e f o r m a t i o n . ) By a p p l i c a t i o n of the theory of OL, the dependence of the d e c r e m e n t and the modulus defect on the degree of deformation, are not too large,

at least with d e f o r m a t i o n s which

is explained by a linear increase

in the d i s l o c a t i o n density

and a d e c r e a s e of the loop length with increasing cold work.

We wish to thank Prof. R o l l w a g e n for the interest he has shown in this study,

the D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t

as Prof.

Schwink for his stimulus.

for its financial

support,

as well

534

AMPLITUDEINDEPENDENT INTERNAL FRICTION IN COPPER (1)

Vol.

7, No.

REFERENCES 1.

A. Granato and K. L~cke, J. Appl. Phys. 27, 583 (1956)

2.

K. Bei~ner and E. Billet, Scripta Met., this issue

3.

A. W. Lawson, Phys. Rev. 60, 330 (1941) R.R. IIasiguti and T. Hirai, J. Appl. Phys. 22, 1084 (1951) J. Weertman and J.S. Koehler, J. Appl. Phys. 24, 624 (1953) A. Hikata, R. Truell, A. Granato, B. Chick, and K. LNcke, J. Appl. Phys. 27, 396 (1956) A. J. Brouwer and C. Groenenboom-Eygelaar,

Acta Met. 15, 1597 (1967)

A. van den Beukel and C. Brouwer, Phil.Mag. 17, 453 (1968) 4.

J. E. Bailey, Phil. Mag. ~, 223 (1963)

5.

U. Essmann, M. Rapp, and M. Wilkens, Acta Met. 16, 1275 (1968)

6.

S. G. McDonald and N.F. Fiore, Scripta Met. ~, 135 (1970)

S