EQUATION OF MOTION OF A SCREW DISLOCATION IN A VISCO-ELASTIC CONTINUUM

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EQUATION OF MOTION OF A SCREW

DISLOCATION IN A VISCO-ELASTIC CONTINUUM

J. Schlipf

To cite this version:

J. Schlipf.

EQUATION OF MOTION OF A SCREW DISLOCATION IN A

JOURNAL DE PHYSIQUE

Colloque CIO, supplément au n012, Tome 46, décembre 1985 page CIO-155

EQUATION OF MOTION CONTINUUM

O F . A SCREW DISLOCATION A VISCO-ELASTIC

J. SCHLIPF

Institut

für

Allgemine Metallkunde und Metallphysik, T e c h . Hochschule, 0-5100 Aachen, F.R.G.

Abstract

-

S t a r t i n g from a Lagrangean formalism f o r a l i n e a r visco-elastic

continuum t h e d i s s i p a t i v e forces a r e eliminated by an appropriate trans-

formation. The equation of motion i s then obtained by variation of t h e

action integral i n terms of Green's functions. The r e s u l t can be expressed

as a s t r i n g equation with a damping t e m . The contribution of t h e shear

viscosity t o t h e damping constant i s calculated on the basis of a Grüneisen

constant f o r shear. Effective mass and l i n e tension a r e a l s o given.

1

-

INTRODUCTION

In several papers Brailsford

/ T /

has shown how the motion of dislocations i s impeded

by phonon and electron v i s c o s i t i e s . He decomposed t h e bulk and t h e dislocation dis-

placements i n t o i t s Fourier components and calculated the damping as a function of

t h e wave v e c t o r k . Bross /2/ and Stenzel

/3/

on t h e other hand proposed t o s t a r t

from the equation of motion of an e l a s t i c continuum and derived an equation of

motion f o r the screw dislocation, using Green's functions. In the following i t i s

shown how t h i s procedure can be implemented t o include viscous e f f e c t s .

I I

-

DYNAMICS

OF THE

VISCO-ELASTIC CONTINUUM

Starting from theLaqangeanenergy density

L =

T-V

and t h e d i s s i p a t i v e function $

t h e equations of motion f o r a visco-elastic continuum read:

a

a~

a a~

-

a

aili

- - + - - -

a t

a i l

ax. a s

axj

abG

( 1

1

J

l j

1

j

Here s l and

jl

denote the components of the t o t a l displacement and veloci t y , respect-

ively, of a volume element a t (x. t ) , ~ l j

and Blj represent e l a s t i c and plastic de-

formation, respectively,and

BG

-:

ajs1. Finally,

T = (1/2) P

i181

;

1

j

V =

(1/2)Cl

jmn €1 j Emn; @ =

(1/2)nl

j m n a j i l a m i n ,

and

p

denotes mass density. Cl

jmn

and nljmn are tensor of e l a s t i c i t y and viscosity, respectively.

CIO-156

JOURNAL

D E PHYSIQUE

I n t h e absence o f d i s s i p a t i o n (1) can be o b t a i n e d by m i n i m i z i n g t h e a c t i o n i n t e g r a l :

Using t h e dyadic Green's f u n c t i o n G i j ! - x ' , t - t ' ) Stenzel /3/ has shown how i n t h i s case an equation o f motion can be d e r i v e a from ( 1 ) and ( 2 ) . I n t h e present work h i s procedure i s extended t o t h e case o f a viscous medium.

Since (2) holds f o r energy conserving systems o n l y i t cannot be a p p l i e d d i r e c t l y i n t h e case o f a viscous medium. We t h e r e f o r e seek a t r a n s f o r m a t i o n w i t h r e s p e c t t o t i m e such t h a t t h e r i g h t - h a n d s i d e o f (1

1

vanishes. With t replaced by Q ( t ) , and $ ( t )

=

d e / d t , ( 1 ) gives us

Thus, i f i t i s p o s s i b l e t o f i n d a t r a n s f o r m a t i o n Q ( t ) such t h a t t h e sum o f t h e f i r s t o r d e r tenns

a/aQ

vanishes, t h e problem w i l l be reduced t o t h e d i s s i p a t i o n - f r e e case:

Now ( 4 ) can indeed be s a t i s f i e d , i f s i i s decomposed i n t o p l a n e waves S ( X ) = C

C(k,?);

s; = ul e x p ( i k 5 )

-

-

(5)

and i f

$/$I

- a (6)

i s independent of t. I n t h e isotropie continuum w i t h q,g = l o n g i t u d i n a l and shear v i s c i s o t y , r e s p e c t i v e l y , we have f o r

-

l o n g i t u d i n a l modes: a = kZ(rl + ~ c ) / P (7a) 2

t r a n s v e r s e modes :

cx

= k S / P . (7b)

I n t r o d u c i ng t h e o p e r a t o r

(1)

may

be broken down t o

which permits a p p l i c a t i o n o f t h e minimum a c t i o n p r i n c i p l e (2). II- SCREW DISLOCATIONS

L e t t h e screw d i s l o c a t i o n extend over -L/2 5 x3 5 L/Z and move i n t h e XI, x3 plane,

where i t i s represented by y(x3,t) w i t h l y l << L. I n o r d e r t o a v o i d divergencies a t

XI = y we w r i t e i t s p l a s t i c d i s t o r t i o n /3/:

where

J

Here,

Ko; KI,

K114

are modified Bessel functions of the second kind,

u;

v

e l a s t i c

constants, and

CT; CL

transverse and longitudinal sound velocity, respectively.

S'nce the screw dislocation produces only shear, a ( k )

according t o (7b) i s given by

kt<(kl)/p. Brailsford /1/ has given.expressions f o r the longitudinal viscosity n(k).

He has l i s t e d four effects: s c a t t e r i n g of phonons and electrons (high frequency

modes) and t h e n o e l a s t i

c

effects due t o phonons and electrons (low f requency modes).

The effects considered by Brailsford a r e al1 due t o density changes, they cannot be

applied t o screw dislocations.

Brai l s f o r d ' s calculation of the phonon e f f e c t s i s based on a Grüneisen constant

L Y

(k) cv(k)

a

in

w(k)

Y =

;

Y(&)=

a a

( 1 5 )

c

c v ( k )

where c v ( i ) i s the contribution t o t h e s p e c i f i c heat of t h e wave vector

k,

w(k)

a r e

the eigenfrequencies, and

A

i s volume dilatation. Bond changes e n t a i l i n g frequency

changes a r e produced not only by volume d i l a t a t i o n ,

b u t

also by shear. Therefore, a

Grüneisen constant may be defined also f o r shear:

where the effective o r octahedral shear

has been introduced, and

€ 1 ,

€2,

€3

a r e the principal s t r a i n s . The phonon s c a t t e r i n g

due t o screw dislocations may then be calculated in the same way as f o r edge dislo-

cations

/1/,

yielding f o r the damping constant

JOURNAL

DE

PHYSIQUE l + v L 1

c

= %

[- l n - + Z ( ~ E

+

3 1 n 2 )

+

2 3/4] ro CE = 0.577... ( E u l e r ' s constant). I V

-

DISCUSSION

The present t h e o r y c o n f i n n s t h e s t r i n g equation f o r t h e motion o f t h e d i s l o c a t i o n . It contains a damping t e n n which, f o r screw d i s l o c a t i o n s , i s s o l e l y due t o phonon s c a t t e r i n g . The expressions (1 9), (20), a r e e s s e n t i a l l y i d e n t i c a l w i t h those given by Stenzel /3/. Also t h e formula f o r B shows o n l y minor, though s t i l l noteworthy, d i f f e r e n c e s w i t h t h e corresponding t e n n i n B r a i l s f o r d ' s t h e o r y f o r edge d i s l o c a t i o n s :

I n t h e present treatment t h e f a c t o r cf/$ disappears i n t h e damping constant, w h i l e t h e w i d t h ro o f t h e d i s l o c a t i o n core comes i n .

The appearance o f ro i n (18) causes B t o d i v e r g e as ro+ O which seems p l a u s i b l e , as i t i s a l s o found i n A and C.

REFERENCES

/1/ A.D. B r a i l s f o r d , Proc. 5, ICIFUAS, (1973) Vol. II, 1. /2/ H. Bross, phys. s t a t . s o l . 5 (1964) 329.

/3/ G. Stenzel, phys. s t a t . sol,

-

34 (1969) 351, 365.

ACKNOWLEDGEMENT