THE THEORY OF STRAIN AMPLITUDE DEPENDENT DISLOCATION DAMPING IN THE PRESENCE OF UNIFORM POINT DEFECT DRAGGING

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THE THEORY OF STRAIN AMPLITUDE

DEPENDENT DISLOCATION DAMPING IN THE PRESENCE OF UNIFORM POINT DEFECT

DRAGGING

T. Ogurtani, A. Seeger

To cite this version:

T. Ogurtani, A. Seeger. THE THEORY OF STRAIN AMPLITUDE DEPENDENT DISLOCA-

TION DAMPING IN THE PRESENCE OF UNIFORM POINT DEFECT DRAGGING. Journal de

Physique Colloques, 1983, 44 (C9), pp.C9-619-C9-625. �10.1051/jphyscol:1983993�. �jpa-00223443�

JOURNAL DE PHYSIQUE

Colloque

C9,

suppl6ment

au

n012, T o m e

44,

d6cembre 1983 page C9-619

THE THEORY OF STRAIN AMPLITUDE DEPENDENT DISLOCATION DAMPING IN THE PRESENCE OF UNIFORM POINT DEFECT DRAGGING

T.O. Ogurtani and A + Seeger

Max-PZanek-Institut far MetaZZforschung, Institut f a Physik, Stuttgart, F.R.G.

and MiddZe East TechnicaZ University, Ankara, !Turkey

Abstract.- The strain-amplitude and frequency-dependent internal friction due to movements of dislocations in the presence of uniform point defect draggiv,~ and randomly distributed, weak pinning obstacles is investigated using the string model. By making use of computer simulations with the ca- tastrophic breakaway distribution function and uniform point defect dragg- ing, the internal friction coefficient is obtained as a function of the stress amplitude, homologous temperature, driving frequency, and densities of the dragging point defects and the weak pinning obstacles.

1. Introduction.- The effect of large stress amplitudes on the decrement associated with the dislocation damping phenomena has been discussed with partial success in a model based upon the mechanical depinning of dislocations from weak pinning-point obstacles /1,2/. In the original theory of Granato and Lucke (GL) /2/ for the hyste- retic damping, not only were the effects of dragging point defects neglected but also some analytical approximations were introduced which limited the applicability of the model even for moderate stress amplitudes. However, in contrast to common arguments in the literature /3/ one can easily include the effects of finite temperature, as we will demonstrate later, by adopting temperature-dependent pinning forces.

The most important shortcoming of the classical string treatment /2/ is the omission of the frequency dependent part of the logarithmic decrement which generates critical situations in both the very small and the large stress amplitude regions. Also, the statistical loop-length distribution function employed in the GL theory yields an underestimation of the decrement at stress amplitudes well below the peak value of the depinning process. At higher stress values, the situation becomes even worse according to our recent, extensive computer experiments

/4/.

In order to obtain a self-consistent theory of the depinning process, which is valid for the whole range of stress amplitudes, the statistical distribution function should be kept as exact as possible and should be utilized precisely in numerical computations.

The thermally activated breakaway of dislocations has been extensively studied by

Teutonico, Granato, and Lucke /3/. Similarly, Koiwa and Hasiguti

/5/,

Peguin arid

Birnbaum

/6/,

and Lucke, Granato, and Teutonico /7/ tried to find approximate solu-

tions to the problem without introducing the dragging defects (Simpson and Sosin

/8, 9/ )

explicitly into the string model, in order to study the influence of temperature,

applied stress amplitude, dislocation length or the number of pinning obstacles

/ l o / . Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983993

C9-620 JOURNAL DE PHYSIQUE

Recently, Schwarz /11/ performed extensive computer simulation studies using the string model, which is strictly valid only crystals with low Peierls bariers /12,13/, in a random distribution of weak obstacles and including inertial and viscous forces.

However, the effect of point defect dragging and consequently the dependence of the decrement on the frequency is completely neglected or at least underestimated.

2.- Statistical Distribution Function.- In this section as far as the main statisti- cal aspect of the distribution function of N.(R,~) is concerned we will make use of the catastrophic breakaway model which was originally proposed by Koehler /I/, and

J

highly refined by Granato and Lucke /2/. Where, o is the instantaneous value of the resolved shear stress acting on the dislocation line. In the general formulation of the problem, especially for numerical solutions, in contrast to GL we will keep the exact form of the probability, M, that a catastrophe has occurred on the network length, as

rather than the expression used by GL as a good approximation for the early stages of the breakaway process,

i

.e. M = ^n(q

+

1) exp(-q), where q=L/Lc, L is the criti- cal length, and LC is the mean length between the weak pinning points. Here, n is the number of weak pinning points on the network length. Two distinct types of pin- ning points are assumed to be situated along the dislocation line. The first type are the network pinning points which have infinite breakaway strength, and are uni- formly distributed along the dislocation with an average separation, LN. The second type of pinning points are randomly arranged along the dislocation segments and ex- hibit a finite breakaway strength with a maximum pinning force, f , . When the force exerted by the dislocation line on this pinning point is larger than fm, the so-call- ed catastrophic breakaway process starts and continues until the whole dislocation segment, between two strongly 'pinned network points, has broken away completely. Using the statistical arguments along these lines, as advocated by GL, one obtains the following distribution function from eq.(l) which has a wider range of validity, but still recognizes that the loop lengths are statistically independent /4/:

N1(gy

WY) =

A/LN

( 1

- (1 - ⁽¹ - ⁽¹

⁺

q) e~~(-~))~})6(~- L ~ ) , L

< L < ( 3 )

and

where

O(z)

is the Heavyside function, y= LN/Lc,and the average value of n is given by n

=

y- 1. Furthermore, we have N2(&,q)

=

N1 (a, qo) when O<R<W . ^The

critical breakaway length may be given by L =fm/o , where the exact nature of the

proportionality constant depends upon the specific model. According to our analysis:

2

q

=

Tc/o , ^where rc is given by

TT

fm/4aL, as a first order approximation. Similar-

3.- Dislocation Oscillations Under the Effect of Uniform Point Defect Drag.- In addi- tion to the immobile point defects exhibiting a finite breakaway strength we will also consider those point defects which are highly mobile due to the thermal fluctuations, and exert viscous drag effect on the moving dislocation segment. According to the breakaway statistics adopted in the present work, the pinning points due to the immo- bile impurities represent rigid boundary conditions below a certain critical resolved shear stress which can be calculated from a given value of fm in terms of the ad- joinning loop lengths. Therefore, the unified dislocation damping theory associated with the point defect dragging developed by Ogurtani /14/ is strictly valid in the calculation of the displacement S(R,t) of the dislocation line segment up to the critical resolved stress level, beyond which the catastrophic breakaway or depinning process starts and continues until the whole dislocation segment in question, between two network points, has broken away. Hence, we can write the following expression:

where R

< L,

and similarly for the dislocation strain E ~ ( R , ~ ) one can deduce:

where

a2 = i

C/ w(iw

+

d)A, and d= B/A. In the above exact expression, A is the effective mass per unit length of the dislocation segment, C is the :ine tension, which is assumed to be constant along the dislocation line, and B is the effective damping constant associated with uniform drag. B is given by /4,14/

where, Bo is the.viscous damping constant (background damping) in the absence of the mobile defects, and Bd

i s

the damping constant for each of the Na dragging point defects along the dislocation loop of length, R. In the case of a uniform distribu- tion of the mobile defects, we can made the following substitutions: Ed

=

kT/Da and (Na

+ I

) / a

=

nay where Da is the diffusivity of the mobile dragging defects in the d immediate vicinity of the dislocation line, and d

na is the mean linear density of

these defects at the dislocation segment. The temperature dependence of na is very d

critical in character and shows a Fermi-Dirac type functional behaviour /4/ with low

temperature, site saturation. The relationship

( 6 )

can also be written in terms of

the instantaneous value of the shear stress, showing explicitly two distinct branches

depending upon whether it is increasing or decreasing, i.e. a minus or plus sign in

front of the second term, respectively

/ I

41

JOURNAL DE PHYSIQUE

2 2

where fig =

nB,N

(!L/L,)~ y2, and

nBSN

= uB LN /n C which emphasizes t h e l o o p l e n g t h dependence o f

nB

expl i c i t l y .

4.- S t r a i n Amplitude Dependent Damping i n t h e Presence o f P o i n t D e f e c t Dragging.- The l o g a r i t h m i c decrement can be o b t a i n e d i n t h e f o l l o w i n g form:

where AW(ao) i s t h e energy d i s s i p a t i o n per u n i t volume per c y c l e o f t h e sample which may be w r i t t e n as:

4 ⁰ Go

AW(oo) =

-

2 {

{

ElYd(a) do

- {

~ ~ ~do ~

1 .

( 0 ) (10) The t o t a l d i s l o c a t i o n s t r a i n f o r a given instantaneous value o f t h e r e s o l v e d shear s t r e s s

u,

can be w r i t t e n as

m

E ~ , ~ ( u ) =

{

E ~ , ~ ( R , o ) Nj(R,a) dk, j = 1,2. (11) I n eq.(9)

so

i s t h e amplitude o f t h e u n i a x i a l a p p l i e d stress, and E(e,+) i s t h e YoungS modulus o f t h e s i n g l e c r y s t a l ,

+,e

a r e t h e angles which r e l a t e a p p l i e d s t r e s s t o t h e r e s o l v e d shear s t r e s s i n t h e s l i p plane and i n t h e s l i p d i r e c t i o n , r e s p e c t i v e l y . R(e,+) i s Schmid's f a c t o r which i s given by cose sine cos4

.

5.- Discussion.- I n F i g u r e ( 1 ) t h e e x c l u s i v e l y amplitude dependent p a r t o f t h e l o - g a r i t h m i c decrement f o r a s i n g l e c r y s t a l i s p l o t t e d (QB= 0) versus t h e normalized u n i a x i a l a p p l i e d s t r e s s amp1 i t u d e

So/Tc f o r v a r i o u s values o f t h e number o f weak p i n n i n g obstacles, u s i n g t h e p l o t t e r f a c i l i t i e s o f an HP-9821A minicomputer. I n t h i s example we have s e l e c t e d R = 1/2, which corresponds t o t h e best o r i e n t a t i o n o f t h e a c t i v e d i s l o c a t i o n s l i p system w i t h r e s p e c t t o t h e u n i a x i a l s t r e s s system. There a r e two important f e a t u r e s associated w i t h these p l o t s , f i r s t l y t h e f a c t t h a t t h e maximum i n t h e decrement ( h y s t e r e t i c i n o r i g i n ) occurs a t about C o n c = 1.5 regardless t h e value o f y. Secondly, t h e n o n l i n e a r behaviour o f t h e decrement w i t h r e s p e c t t o i t s dependence on t h e o b s t a c l e d e n s i t y extends up t o t h e v a l u e o f y = 50. The most i n t e - r e s t i n g f e a t u r e o f t h e whole d i s l o c a t i o n damping phenomenon i n t h e presence of t h e dragging p o i n t d e f e c t s p l u s t h e weak obstacles can o n l y be deduced from an exact nume- r i c a l s o l u t i o n /4/.

I n F i g u r e ( 2 ) t h e l o g a r i t h m i c decrement obtained from t h e r i g o r o u s numerical s o l u t i o n u s i n g eqs.(3-11) i s presented as a f u n c t i o n o f t h e normalized s t r e s s amplitude f o r v a r i o u s values o f t h e normalized frequency RB. This p l o t i n d i c a t e s t h e e x i s t e n c e o f

t h r e e d i s t i n c t stages as f a r as t h e s t r e s s amplitude dependence i s concerned: t h e s t r e s s amplitude independent low s t r e s s l e v e l , t h e depinning c o n t r o l e d t r a n s i t i o n r e - g i o n a t medium s t r e s s l e v e l s and f i n a l l y , t h e s t r e s s amplitude i n s e n s i t i v e stage a t l a r g e s t r e s s l e v e l s . The maximum i n t h e decrement occurs a t e x a c t l y QB= 0.1 i n t h e s t r e s s amplitude independent stage. I n t h e depinning stage t h e r e i s a continuous t r a n s - f o r m a t i o n from t h i s peak i n t o another one w i t h a maximum normalized frequency o f QBYN=1, which corresponds t o RB = l / y

.

Closer i n s p e c t i o n c l e a r l y r e v e a l s t h a t a t t h e end o f t h e depinning process a l l t h e d i s l o c a t i o n segments w i l l have l e n g t h

LN

g i v i n g a sharp r e l a x a t i o n maximum a t e x a c t l y

RBYN = 1 due t o t h e D e l t a d i s t r i b u t i o n f u n c t i o n of t h e segment l e n g t h s /15/.

The temperature dependence o f t h e amplitude dependent as w e l l as frequency dependent d i s l o c a t i o n damping c o u l d a r i s e through t h r e e d i s t i n c t and i m p o r t a n t f a c t o r s according t o t h e present model c a l c u l a t i o n s . The f i r s t f a c t o r i s t h e temperature dependence o f t h e drag constant B. The second f a c t o r which a l s o cause v a r i a t i o n s i n t h e decrement w i t h temperature i s due t o t h e c o n c e n t r a t i o n o f t h e weak obstacles along t h e d i s l o c a - t i o n l i n e through t h e parameter y . T h i s c o n c e n t r a t i o n a l s o shows a Langmuir-type ad- s o r p t i o n isotherm. The t h i r d b u t t h e most important c o n t r i b u t i o n t o t h e temperature dependence comes from

rc,

t h e depinning stress, through f,, t h e maximum depinning f o r c e . The temperature dependence o f fm may be represented by t h e f o l l o w i n g ad hoc expression which i n d i c a t e s t h a t t h e thermal a c t i v a t i o n o f the depinning process below a c e r t a i n c h a r a c t e r i s t i c temperature w i l l be a l m o s t l y f r o z e n - i n . Hence:

where

H!:

i s t h e depinning a c t i v a t i o n enthalpy, f i i s t h e depinning f o r c e a t zero absolute temperature, and m

fm i s t h e same f o r c e a t i n f i n i t e temperature.

I n F i g u r e ( 3 ) t h e l o g a r i t h m i c decrement obtained from t h e numerical s o l u t i o n i s p l o t t e d as a f u n c t i o n o f t h e homologous temperature, T/TM, f o r v a r i o u s values o f t h e s t r e s s amplitude on a double l o g a r i t h m i c scale. T h i s f i g u r e c l e a r l y r e v e a l s t h e existence of two d i s t i n c t peaks. The f i r s t peak which appears a t t h e low temperature s i d e i s due t o t h e dragging p o i n t defects. I t has t h e n a t u r e o f viscous damping as evidenced by i t s s t r o n g frequency dependence and s t r e s s amp1 i tude independence. The second peak which i s r a t h e r skew on t h e h i g h temperature shoulder shows s t r o n g s t r e s s amplitude dependence o f i t s peak p o s i t i o n , and complete i n s e n s i v i t y w i t h r e s p e c t t o t h e v a r i a - t i o n s i n t h e e x c i t a t i o n frequency. T h i s second peak i s c l o s e l y r e l a t e d t o t h e depin- n i n g process o f t h e d i s l o c a t i o n segment from t h e weak obstacles.

REFERENCES

1. J.S. Koehler, I n I m p e r f e c t i o n s i n N e a r l y P e r f e c t C r y s t a l s ( W . Shockley, J.H.

Hollomon; R. Mauerer and F. S e i t z , ed.) J. Wiley, New York, (1952).

2. A. V. Granato and K. Lucke; J. Appl. Phys., 27, 583: 789 (1956).

3. L . J. Teutonico, A. V. Granato, and K. Lucke, J. Appl. Phys., 35, 220 (1964).

C9-624 JOURNAL DE PHYSIQUE

T.O. Ogurtani and A. Seeger, "Theory o f Strain-Amplitude-Dependent D i s l o c a t i o n Damping i n t h e Presence o f P o i n t Defect Dragging", accepted f o r p u b l i c a t i o n i n Phys. Rev. B. ( T e n t a t i v e issue- December, 1983).

M. Koiwa and R.R. H a s i g u t i , Acta Met., 13, 1219 (1965).

P. Peguin and H.K. Birnbaum, J. Appl. Phys., 39, 4428 (1968).

K. Lucke, A.V. Granato and L. Teutonico, J. Appl. Phys., 39, 5181 (1968).

H. M. Simpson and A. Sosin, Phys. Rev. 5B, 1382 (1972).

H. M. Simpson and A. Sosin, and D. F. Johnson, Phys. Rev. 5B, 1393 (1972).

R. Klam, H. Schultz, and H. E. Schaefer, Acta Met., 28, 259 (1980).

R. B. Schwarz, Acta Met., 29, 311 (1981).

A. Seeger, J. De Phys. C o l l . C.5, Suppl. 10, 201 (1981).

G. Fantozzi, C. Esnouf, W. Benoit, and I . G. R i t c h i e , Prog. i n Mat. Sci.,27, 311 (1982).

T. 0. Ogurtani, Phys. Rev. B, 21, 4373 (1980).

T. 0. Ogurtani, J. De Phys. C o l l . C.5, Suppl. 10, 235 (1981).

ACKNOWLEDGEMENTS

The authors wish t o thank Professor H. S c h u l t z f o r v a l u a b l e discussions. Thanks a r e a l s o due Dr. K. D i f f e r t f o r t h e p r e l i m i n a r y computer o r i e n t a t i o n s t u d i e s .

F I G U R E C A P T I O N S

1. The e x c l u s i v e l y amplitude dependent p a r t o f t h e l o g a r i t h m i c decrement i s p l o t t e d w i t h r e s p e c t t o t h e normalized a p p l i e d s t r e s s amplitude f o r v a r i o u s values o f t h e number o f weak p i n n i n g p o i n t s .

2. The frequency dependent p a r t o f t h e l o g a r i t h m i c decrement

is

p l o t t e d w i t h r e s p e c t t o t h e normalized a p p l i e d s t r e s s amplitude f o r t h e v a r i o u s values o f t h e n o r m a l i - zed frequency. The t o t a l decrements a r e obtained according t o t h e numerical i n - t e g r a t i o n .

3 . The t o t a l l o g a r i t h m i c decrement according t o t h e r i g o r o u s t h e o r y i s p l o t t e d as a f u n c t i o n o f t h e homologous temperature T/TM, f o r v a r i o u s values o f t h e normalized s t r e s s amplitude.

NORMALIZED STRESS AMPLITUDE ; ( 0 /TC) 6

I-

6

6

^FIGURE ( t D R U E U R T R N I

a

W 0

.

P! ^J SXTRL B E S T PRlENTRTiUN

a

T:

.

6

Z I R I = 0 . 5

..--...

. . . .-..

^>

. m . m s :

...:

^{- 50} P R U G R R M 3 I

. . . . ..-... . . . ...

. .

. . ...

... _...-.-

⁼

...

^,O

... ... ... ... ... ... ...

.

.

. .

^.C

-... "... ....

. . . . .

_{y ' 5}

.-. ... ... ...

^*

..

...

" . -... ..-... .... ....*

- . . -... -.... '".--...

,"-...--.a,_

. . ..* ---...

"

----.-.._ --.-

-------*,

. .

s l

STRES5/B-STRESS

.I. .

I

R M C L I T Y D E R N D F R K Q U L N C Y D E F L N D E N I D E C S E M E N T I D E U R T R N I I S E C G E R E X R C T S O L U T I O N

ICG ( NOFMALIZED STRESS ( Do/ Tc 1 AMPLITUDE )

R M P L I T U D E R N D F R E Q U E N C Y D E P E N D E N T D E C R E M E N T

I E X R C T P O L U T 1 ON

FIGURE ( 3 )

0

DATA :

- I R / = 0.5

H . / k T = 2 ^{R R R}

- I 1 H

E, / k T = 2 1.b M H Z / k T M = 1 - 2

r:/rZ = l o 3 D

-

^{= 1}

.

^X' = 1 E - 4 y = 2 0

-

^q "D" DWIGGING PEAK

"R" DePINNING PEAK

-

^L

- 2