In direct mobility experiments in single crystals, dislocation velocity is studied as a function of stress by the application of short-duration stress pulses. The stress pulse consists of a loading wave, followed microseconds later, by an unloading wave. At high velocities, dislocation inertia effects become important if the dislocation damping force is a decreasing function of dislocation velocity. In general, the magnitude of this force can be determined only if the relative velocity between the applied stress wave and the dislocation is considered

Jassby, K. M.

Vreeland, T., Jr.

UNT (University of North Texas) Digital Library

the Measurement of Dislocation Dam in%,:!.:. -< dL$i * Forces at  High Dislocation Velocity K. M. Jassby and T. Vreeland, Jr. w . M. Keck Laboratories, California Institute of Technology Pasadena, California ABSTRACT In direct mobility experiments in single crystals, dislocation velocity i s  studied as  a function of s t ress  by the application of short-duration s t ress  pulses. The stress pulse consists of a loading wave, followed microseconds later, by an unloading wave. At high velocities, dislocation inertia effects become important only if the dislocation damping force is a decreasing function of velocity. In general, the magnitude of this force can be determined only i f  the relative velocity between the applied stress wave and the dislocation is considered. Then is no objettion from the patent standpoint to the publication or disreminatiah of this document. * w. , This work was supported by the U. S. Atomic Energy Commission I I / / I - N D t l C E  Thb -port was prepared u, an account of work seommd by the United States Government. Neither the Unit& &tea nor the United States Atomic Energy COmmislion, Bar uy -f thah employees, nor any of their contractors, airsrontractorl), or thek emplayens2 Mew any WBmntY, expms or implied, or assumes MY h n l  UaMlity or responsibility for the accprwy, corn- pletenea or u8efuIne.s~ of my information. appmus ,  product or WOWSB disclol~d, or represents that ib ure would not owmd sigh&. 1 DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. 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INTRODUCTION Experimental work carried out in the las t  few years has confirmed that interaction with thermal phonons and conduction electrons gives r i s e  to the major, source of energy dissipation for moving dislocations in close-packed metals. While phonon damping predominates near the Debye temperature, electron damping i s  strongest in the cryogenic region where strong atomic vibrations a r e  absent. Throughout the temperature 4 range, for dislocation velocities up to about 2x10 cm/sec  the dissipative force i s  directly proportional to dislocation velocity, and for a straight dislocation moving a t  i t s  terminal velocity, i t  i s  equal to the driving force. Therefore, for a unit length of dislocation where B = dislocation damping coefficient, . . v = dislocation velocity, . . 'R = resolved shear s t ress ,  and b = dislocation Burgers vector, ' -4  B i s  the order of 10 cgs. at  room temperature and decreases to - 5  . . about 10 cgs. at 4.2 O K 2 ) .  Hence the s t resses  required to accelerate dislocations to appreciable velocities in high purity,. close-packed metals a r e  considerably smaller than those required in other materials where thermally activated mechanisms predominate (see for example, lithium fluoride, ( 3 )  or  iron. (*I). . . considerable speculation has centered on the question of an upper limit to dislocation velocity. (5) Continuum linear isotropic elasticity theory predicts that the total energy and effective mass  of a dislocation become infinite at  a velocity equal to the shear wave velocity in the ' t ' . ( 6 )  material,  C and therefore this velocity i s  an upper limit for dislocations. However, in an elastic solid, the increase in magnitude of the dislocation s t r e s s  field would make the linear theory invalid a t  velocities greater than about 0. 95Ct.' When the discrete nature of the crystal lattice and, hence, wave dispersion, a r e  considered, then a dislocation moving more  quickly than the minimum sound velocity experiences an additional retarding force from the medium, which i s  very eensitive to the structure of the dislocation core. (7) Also several atomistic dislocation models have been proposed in which the dislocation super sonic energy does not diverge at  C but rather the lattice can support t ' dis1ocatio.n motion. (8)' Various theoretical predictions concerning dislocation velocities can be tested by experimental techniques. In the usual direct method for I measuring dislocation velocity a s  a function of resolved shear s t ress ,  a s t r e s s  pulse of known magnitude and duration i s  passed through a crystal and the distances individual dislocations move a r e  observed. The s t r e s s  pulse produces a resolved shear s t ress  a t  the dislocation, given by rR( t ) ,  where t i s  equal to time. (Note &hat the position of &he dislocation will be changing, so ,  that determination of T ( t)  may be complicated. ) Dislocation R I displacement i s  equal to the time integral of the dislocation velocity, that i s  A relation between dislocation velocity and resolved shear s t r e s s  i s  postulated, and substituted into Eq, 2, so that ' . represents 'the.displacement of the dislocation provided Eq. 3 i s  valid. When d '  is compared to the actual displacement d, agreement implies , . that Eq. 3 i s  indeed correct,  and the damping force on the dislocation i s  deduced. lmpiicit: in Eq. 3 is the assumption that inertial effects a r e  unimportant, that is dislocation velocity does not depend explicitly on time. In this paper, we shall discuss under what experimental conditions this a s  sumption remains valid. I t  is re-called that the usual method of s t ress  pulse application i s  by means of a loading wave followed later  by an unloading wave. If the direction of s t ress  wave propagation has a component in the direction of dislocation motion, then the relative velocities of the loading and unloading waves with respect to the dislocation must be considered to accurately determine ~ ~ ( t ) .  In this paper we shall see that this consideration is particularly important in the high-velocity region. DISLOCATION INERTIA AND ' . ,. ACCELERATION TIME CONSTANT In the general case, dislocation velocity i s  dependent explicitly . . . . on both resolved sheai  s t ress ,  T and time t. The functional form R ' for this dependence i s  derived from the equation of motion for a long, straight dislocation d - (mv) = ~ ~ ( t ) b  - B(v) d t (5) B(v) = dissipative force which i s  some function of dislocation velocity and m = effective mass  per unit length of the dislocation. m has been ca1,culated for both edge and screw dislocations on the basis of isotropic elasticity theory, and for a screw dislocation in uniform motion is given, by i t s - r e s t  value divided by a velocity contraction factor. (6) where P = density of t h e  medium R = outer cut -off radius r = inner cut -off radius 0 . . A similar r e i ' d t  holds for an edge dislocation. For non-uniform motion, any change in m with velocity contains an explicit time dependence through the  logarithmic term,  which expresses the period rsquired for elastic .. . waves denoting a velocity change to radiate from the dislocation core. (9 ) I t  i s  useful to consider the motion of a dislocation, initially moving &th uniform velocity v caused by a sudden small change in applied 0' s t ress .  Then the response to an arbitrary s t r e s s  change can be found by . . integration of a ser ies  of impulsive changes in s t ress .  In the limit of 'small velocities, v < < Ct, m reduces to i t s  res t  value, m and B(v) 0' i s  equal to Bv. Eq. 5, with ~ ~ ( t )  equal to T integrates quite simply to R - 10 The acceleration time constant, mo/B, i s  of the order of 10 to . . l o - '  'sec a t  room temperature for close-packed metals and increases with decreasing temperature. In an explicit case, copper, mo/B increases f rom 5 x l 0 ' ~ ~ s e c  a t  2 9 6 O ~  to 9 ~ 1 0 - ~ ~ s e c  a t  4 .2 '~ .  (10) However, even this lat ter  value for copper i s  insignificant with respect -6  to the r i se  time of a stress*wave in a mobility expeiiinent (> - 10 sec). I .  Hence inertia 'effects a r e  negligible in the low velocity region, and dislocation velocity, to any measureable degree, will be in phase with applied stress.  ' Consider, now, the high velocity region, where the velocity contraction . , factor in Eq. 6 becomes important. Even a t  cryogenic temperatures, provided B does not decrease with velocity, m / B  reaches a maximum of only -- -about 1 0 ' ~ s e c & t h  v in excess of 0.99 Ct, or a t  a considerably larger  velocity than the limit at  which isotropic elastic theory breaks down. Therefore only -- if B decreases with velocity, and in the velocity region --7 : where it  is reduced so that m / B  > 10 sec ( < l o %  - of the r i se  time o f  a wave), can inertial effects, and hence the explicit time dependence of v, be . .  . . important. [~ne r t i a l  effects can always be ignored for short rise-time, long'duration s t r e s s  pulses, where time a t  a constant s t ress  i s  much greater than the r i s e  time of loading and unloading wavefronts. Unfortunately, this experimental' condition is unfeasible a t  high velocities', for considering a - 6  r i s e  time of t h e  order of 10 sec, i t  would imply impractical dislocation displacements of the order of one cm. ] DISLOCATION MOTION CAUSED .BY AN IMPINGING STRESS. WAVE In the following, we calculate the stress-t ime history T ( t) ,  for a R long, straight dislocation responding through i ts  motion to an impinging s t r e s s  wave. As an illustrative example in the analysis, we utilize a linear relation between velocity and s t r e s s  (Eq. l ) ,  but any functional form may be treated by the' :simple method developed. A plane s t r e s s  wavefront, moving in a direction x in a crystal i s  . . . . described by its wave velocity, C, and s t ress  gradient a t  any point a . . . ' ' Assuming that the wave is non-dispersive and the crystal perfectly k elastic, we can write T(X, t )  = T ( X  - + Ct) (10) where Ule negative sign refera to a wave travelling in the positive x-axis direction. Consider a 'dislocation slip direction, . X I ,  at  an angle 9 to the ,  direction of propagation of the wavefront, and lying i n t h e  plane (x, x l )  defined by the normal direction y. A long, straight dislocation moves with velocity v i n  the' positive x '  direction (the slip plane i s  defined by the vector x' x 1). Let the coordinate x" be fixed to the dislocation and - parallel to x'. Then the x-coordinate of the dislocation i s  x = cose [x" + $! v(t1)dt '] ,  where v(0) = 0, and T (x", t )  = T (xl'cosO + coseS~v( t l )d t l  -+ C(t) ) vcose But . . a T a7 cose F It = xi It and using Eq. 11, Eq. 13 becomes a~ vcose - at I x 1 1  = 5 (T I )% 1 X Let  the dislocation velocity be related to the applied s t ress  at  x" = 0 by some continuous function . . Substituting Eq. 3 into Eq. 15, the differential relation for.  the s t ress  -time history of the dislocation i s  dT - + . I d t  . - + L f [  Lcose Equation 16 i s  perfectly general in that its. integration . : depends solely on the functional forms chosen for f {  T ]  and the s t ress  gradient in the. shear wave - :; 1x0 In the following we investigate the case of a dislocation accelerating from re s t  under the influence of a rising s t ress  wave, and constrained by geometry to move in the positive x' direction, and subsequently decelerated by an equivalent unloading wave. We choose where k = constant,' ,and the positive sign refers  to the accelerating wave. This corresponds closely to the situation realized in the usual experiment. Finally, we'assume that a ' linear relation between velocity and re.solved shear s t r e s s  holds through the .velocity range, that i s  where 'a  i s  the, shear s t ress  -resolving factor, that i s  T~ = (YT.  An upper limit for k i s  imposed by the requirement that the duration of time, - 4  from res t  to a maximum velocity, be of the order of 10 sec or greater.  - 5 5 For B equal to 10 ' cgs. and a maximurn velocity of 2x1 0 .cm/sec, then 14 2 k m u s t  be about 10 dyneslcm sec or less. For the accelerating dislocation, Eq. 16 ,becomes d T = kdt - absse 1 , . and integrating 7 = abcose BC ( 1  - exp[- BC'  The dislocation velocity ' i s  . . and i t s  displacement, d, i s  found from According to Eq. 21, the dislocation velocity approaches an upper limit C/cose exponentially, at  a rate determined by the time constant BC This limit of velocity i s  obvious, conceptually, f o r  a dislocation crbkcoee ' can experience no change in the strength of a disturbance when it i s  moving with i t s  velocity of propagation, C. 1n'the real problem, the upper limit of dislocation. velocity i s  not necessarily c /case, which increases without limit a s  8: -, rr/a2. The approach we employed here i s  simply an analysis or relative motion and does not include the limitations imposed upon dislocation velocities in s'olids (see Introduction). In an isotropic continuum, . . . . Eqs. 20 to 22 a r k  valid for the region of velocity l ess  than 0. 99 of the shear wave velocity. This, in  turn, imposes an upper limit in time for . . the' integrability of Eq. ' 19. By combining Eqs. 21 and '22, a simple expression for the distance . . travelled by 'the dislocation. in order ... to .lr:each .a ceritain fraction' of the'Wave velocity i s  found. If we let  v/C = F, then To conclude this section we consider a dislocation initially moving with velocity. v in the ,positive xl-direction interacting with an unloading - m wave moving in the negative x-direction. In this case, Eq. 16 reduces to and integrating, v cose m ) e x p ~  - ~ ~ b k c o s e  C BC t'- ( 2 6 )  v c'os0 and cvbkcos0 t l )  - _t d = ) ( I  -  ex^[ -. .,,Bc c 2 cose qbkcos 0 By. combining Eqs. 26 and 27, we a r r ive  a t  an expression for the distance necessary to bring a dislocation to res t  from a fraction F of the shear wave velocity. DISCUSSION We'have treated the most simple case of an advancing plane s t r e s s  wavefront on dislocation motion, that in which - a' 1 i s  constant and at x dislocation velocity i s  proportional to applied shear s t ress .  Any general form for - 1 or stress-dependence of velocity f [ ~ ~ ]  can be handled, at x although closed-form solutions may not be obtainable in all cases. The example above,. however, is sufficient to point out the importance of considering relative motion when v = f{.rR) i s  to be determined experimentally. As a numerical illuetration, we computed the distance required to accelerate a dislocation to some fraction F of the wave velocity and then bring i t  to res t ,  by applying both the approach outlined above, using Eqs. 23 and 28, and the classigal method where relative motion is ignored and B~~ F ~ ,  T h e  two cases a r e  compared in  Fig. 2 the distance is s in~ply  - cvbk for 0 = 0. . (In the limit 0 = n/2, the two converge. ) .The distance computed . . from the f i rs t  approach begins to diverge noticeably from that of the second 'when F reaches about 0.5. The former case gives a result similar to that when relative motion i s  ignored but the velocity-stress relation i s  assumed to be non-linear with the form where 'n i s  a constant greater than zero. (The case n = $ i s  plotted in Fig. 2. ) Therefore i f  the relative velocity between the dislocation and applied s t ress  wave i s  ignored in the experimental situation where dislocation velocity is indeed linearly related to resolved shear s t r e s s  (B  i s  constant), then a non-linear relation similar to- Eq. 29 will be erroneously indicated where the calculated damping coefficient i s  in e r ro r  by a "relativistic" factor of the form REFERENCES T. .Vreeland, Jr.. and K. M. Jassby,  -- Mat. Sci. Engr . ,  7, 95 (1971). -A. Hikata, R. A. Johnson and C. ~lba&n,  Phys. Rev. B.,  &, -- W. G. Johnson and J. J. Gilman, - J. Appl. Phys . ,  30, 1225 (1970). -A. P. L. Turne r  and T. Vreeland, Jr. , Acta Met. , 18, 1225 (1970). J. Weertman and J. Kraft ,  Dislocation Dynamics, McGraw Hill, New York, . . (1968), p.' 609. J. Weertman, Response - of Metals  to High Velocity Deformation, -In te r  science,  (1961 ), p. 205. J. D. Eshelby, Proc.  Phys. Soc., 69, 1013 (1956). -F. R. N. Nabarro,  Theory of Crys ta l  Dislocations,  Oxford Univ. P r e s s ,  (1967) p. 482. J. P. Hir th  and J. Lothe, Theory of Dislocations, McGraw Hill, New York - (1968), p. 156. K. M. Ja s sby  and T. Vreeland, Jr. , " ~ i r e c t  Measurement  of Phonon and Electron Damping of Edge Dislocation Motion in P u r e  Copper 0 a t  4. 2 Kt',  manuscr ipt  .in preparation. . . . DIRECTION OF PROPAGATION OF STRESS WAVE X DISLOCATION LINE DISLOCATION SLIP Dl RECTION Fig. 1 Dislocation gliding in the s l ip  plane defined by the normal vector z ' ' ~  y. The slip direction x' is a t  an angle 8 f r o m  the direction of propaga- tion of the s t r e s s  wave. n I: CURVE A-  RELATIVE MOTION IGNORED CURVE B-RELATIVE MOTION CONSIDERED CURVE C-RELATIVE MOTION IGNORED WITH "HELATIVISTIC" DAMPING COEFFICIENT BNZ. Fig. 2 Distance required to accelerate a dislocation to a fraction I? of the wave velocity and then bring it to rest, by applying a loading stress wave followed by an unloading wave. 

Measurement of Dislocation Damping Forces at High Dislocation Velocity.

Caltech Authors - Main

On the measurement of dislocation damping forces at high dislocation velocity

Revised Acta Met ms. 9/27/71 
CALT-767 -P3 -15 
On the Measurement of Dislocation Damping 
~:c 
Forces at High Dislocation Velocity 
K. M. Jassby and T. Vreeland, Jr. 
W. M. Keck Laboratories, California Institute of Technology 
Pasadena, California 91109 
ABSTRACT 
In direct mobility experiments in single crystals, dislocation velocity 
is studied as a function of stress by the application of short-duration stress 
pulses. The stress pulse consists of a loading wave, followed microseconds 
later, by an unloading wave. At high velocities, dislocation inertia effects 
become important if the dislocation damping force is a decreasing function 
of dislocation velocity. In general, the magnitude of this force can be 
determined only if the relative velocity between the applied stress wave 
and the dislocation is considered. 
,., 
,,,This work was supported by the U.S. Atomic Energy Commission. 
-2-
I. INTRODUCTION 
During the last few years, dislocation mobility experiments have 
been carried out on a number of metals with close-packed atomic structures 
over wide ranges of temperature (from room temperature down to 4. 2°K) 
and velocity (from 0 to approximately 1 /10 of the sound velocity). The 
experimental work has demonstrated that interaction with thermal phonons 
and conduction electrons gives rise to the major source of energy dissipation 
for moving dislocations, and while phonon damping predominates near the 
Debye temperature, electron damping is strongest in the cryogenic region 
where atomic vibrations are weak. The dissipative force was found to be 
directly proportional to dislocation velocity throughout the velocity range 
studied. During steady-state motion, the dissipative force is equal to the 
driving force, and for a unit length of dislocation, is given by the 
familiar expression 
Bv = TRb' ( 1) 
where B = dislocation damping coefficient, 
v = dislocation velocity, 
TR = resolved shear stress on the dislocation, 
and b = dislocation Burgers vector. 
The magnitude of B is the order of l0-4 cgs at room temperature(!) and 
-5 0 (2) decreases to about 10 cgs at 4. 2 K. Hence the stresses required to 
drive dislocations at appreciable velocities in close-packed metallic 
structures are considerably smaller than those required in materials where 
thermally-activated drag mechanisms predominate (see for example; 
-3-
lithium fluorideP) or iron (4 )). 
Considerable speculation has centered on the question of an upper 
limit to dislocation velocity.(S) Continuum linear isotropic elasticity 
theory predicts that the total energy and effective mass of a dislocation 
become infinite at a velocity equal to the shear wave velocity in the 
material, Ct' and that therefore this velocity is an upper limit for 
dislocations. (b) When the discrete nature of .the crystal lattice and, hence, 
wave dispersion, are considered, then a dislocation moving more quickly 
than the minimum sound velocity experiences an additional retarding force 
from the medium, which is very sensitive to the structure of the dislocation 
core. (7 ) Also several atomistic dislocation models have been proposed in -
which the dislocation energy does not diverge at Ct' but rather the lattice 
can support supersonic dislocation motion. (S) 
Various theoretical predictions concerning dislocation velocities can 
be tested by experimental techniques. In the usual direct method for 
measuring dislocation velocity as a function of resolved shear stress, a 
stress pulse of known magnitude and duration is passed through a crystal 
and the distances individual dislocations move are observed. The stress 
pulse produces a resolved shear stress at the dislocation given by ,-R (t), 
where t is equal to time. (Note that the position of the dislocation will be 
changing, so that determination of TR (t) may be complicated.) Dislocation 
displacement is equal to the time integral of the dislocation velocity, or 
d = J v(t)dt • (2) 
A relation between dislocation velocity and resolved shear stress is postulated, 
-4-
(3) 
This expression is substituted into Eq. 2 to give a hypothetical value for 
displacement· 
(4) 
and if d' agrees with the experimentally determined dislocation displacement, 
then the functional form of Eq. 3 is correct. 
Implicit in Eq. 3 is the assumption that inertial effects are not 
important, that is dislocation velocity does not depend explicitly on time 
but rather is in phase with the applied stress. In Section II we shall discuss 
under what experimental conditions this assumption remains valid at high 
dislocation velocity. 
The usual method of stress pulse application is by means of a loading 
wave followed later by an unloading wave. If the direction of stress wave 
propagation has a component in the direction of dislocation motion, then the 
relative velocities of the loading and unloading waves with respect to the 
dislocation must be considered to accurately determine 'T R(t) as used in 
Eq. 4. In Section III we shall see that this step is particularly important 
in the high-velocity region. 
II. DISLOCATION INERTIA AND THE 
ACCELERATION TIME CONSTANT 
In the general case, dislocation velocity is dependent explicitly on 
both resolved shear stress, 'TR' and timet. The functional form for 
this dependence is derived from the equation of motion for a long, straight 
dislocation, 
-5-
d dt (mv) = 'f R (t)b - B(v)v , (5) 
where B (v) = dislocation damping coefficient, which is some 
function of dislocation velocity, 
m = effective mass per unit length of the dislocation. 
The mass m has been calculated for both edge and screw dislocations on the 
basis of isotropic elasticity theory, and for a screw dislocation in uniform 
motion is given by its rest value divided by a velocity contraction factor. (6) 
That is, 
where 
and 
p = density of the medium, 
R = outer cut-of£ radius, 
r = inner cut-of£ radius • 
0 
(6) 
A similar result holds for an edge dislocation. For non-uniform motion, 
any change in m with velocity contains an explicit time dependence through 
the logarithmic term, which expresses the period required for elastic waves 
denoting a velocity change to radiate from the dislocation core. (9) 
It is useful to consider the motion of a dislocation, initially moving 
with uniform velocity v , caused by a sudden small change in applied 
0 
stress. Then the response to an arbitrary stress change can be found by 
integration of a series of impulsive changes in stress. In the limit of 
small velocities, v < < Ct' m reduces to its rest value, m
0
, and B(v)v 
is equal to Bv. Eq. 5, with TR (t) equal to. TR integrates quite simply to 
v = v 0 
-6-
TRb ( B ) + ~ 1 - exp [ - rn t J (7) 
0 
-10 The acceleration time constant, m /B, is of the order of 10 to 
0 
10 -ll sec at room temperature for close-packed metals and increases by 
about one order of magnitude at 4. 2°K. Typically in a dislocation mobility 
experiment, the applied stress increases from 0 to its maximum value in 
. -6 -5 h 1 . . a tlme between 10 and 10 sec, so t at an acce erabon tlme constant 
less than,.._. 10- 7 sec implies that dislocation velocity is effectively in phase 
with the applied stress, as in the case above. 
In the region of large dislocation velocity, the velocity contraction 
factor in the expression form becomes important. However, even at 
cryogenic temperatures, provided that B(v) does not decrease with dislocation 
velocity, m/B(v) remains less than l0- 8sec for vas large as 0. 99C . Hence 
t 
inertial effects, and the explicit time dependence of v, become important only 
if B(v) decreases as the dislocation velocity increases (v < 0. 99 Ct). 
III. DISLOCATION MOTION CAUSED 
BY AN IMPINGING STRESS WAVE 
In the following, we calculate the stress -time history TR (t), for a 
long, straight dislocation responding through its motion to an impinging 
stress wave. As an illustrative example in the analysis, we utilize a linear 
relation between velocity and stress (Eq. 1 ), but any functional form may 
be treated by the simple method developed. 
A plane stress wavefront, moving in a direction~ in a crystal is 
OT described by its wave velocity, C, and stress gradient at any point at I x· 
Assuming that the wave is non-dispersive and the crystal perfectly 
elastic, we can write 
-7-
'T (x, t) = 'T (x + Ct) , 
( 1 0) 
where the negative sign refers to a wave travelling in the positive x-axis 
direction. 
Consider a dislocation slip direction, x', at an angle 9 to the direction 
of propagation of the wavefront, and lying in the plane (x, x') defined by the 
normal direction y (Fig. 1 ). A long, straight dislocation moves with 
velocity v in the positive ~· direction (the slip plane is defined by the vector 
~· x y_). Let the~~~ direction be fixed to the dislocation and parallel to ~·· 
Then the x-coordinate of the dislocation is given by 
t 
x = cos9[x" + Jo v(t')dt'],v(O) = 0. (11) 
From Eq. 1 0 we find 
t 
'T(x",t) = 'T (x"cos9 + coseJ0 v(t')dt' + C(t) (12) 
and 
It · ( 13) 
But 
(14) 
so using Eq. 10, Eq. 13 becomes 
ch = +(vcos9 + 1 ) o'T at lx" - C at lx • ( 15) 
-8-
Let the dislocation velocity be related to the applied stress at x" = 0 
by some continuous function~ v = f[ 'l"}, as in Eq. 3. Substituting 
this function into Eq. 15, the differential relation for the stress-time 
history of the dislocation is 
+.UhJ cose 
-l c 
( 16) 
Eq. 16 is perfectly general in that its integration depends solely on the 
functional forms chosen for f['l"} and the stress gradient in the shear 
0'1" 
wave at lx · 
In the remainder of this section, we investigate the case of a 
dislocation accelerating from rest under the influence of a rising stress 
wave, and constrained by geometry to move in the positive~· direction, 
and subsequently decelerated by an equivalent unloading wave. We choose 
( 1 7) 
where k = constant, and the positive sign refers to the accelerating wave. 
This corresponds closely to the situation realized in the usual experiment. 
Finally, we assume that a linear relation between velocity and resolved 
shear stress holds through the velocity range, that is 
O'b 
v = B '~" (18) 
where a is the shear stress -resolving factor. An upper limit for k 
is imposed by the requirement that the duration of time, from rest to a 
maximum velocity, be of the order of l0- 6 sec or greater. For B equal to 
-9-
l0- 5 cgs and a maximum velocity of 2xl05 cm/sec, then k must be 
14 2 
about 10 dynes/em sec or less. 
For the accelerating dislocation, Eq. 16 becomes 
and integrating 
abTcose J 
BC 
= kdt , 
BC ( abkcose ) 
T = 1 - exp [- BC t] • abcose 
The dislocation velocity is 
v = 
c 
case ( 
abkcose ) 1 - exp [ - B C t] , 
and its displacement, d, is found from 
t 
d = Jo vdt 
(19) 
(20) 
(21) 
=_g_t+ 
case 
( exp [- ab~~se t] - 1 ) • 
(22) 
According to Eq. 21, the dislocation velocity approaches an upper 
limit C/cose exponentially, at a rate determined by the time constant 
BC This limit of velocity is obvious, conceptually, for a dislocation 
abkcose · 
can experience no change in the strength of a disturbance when it is 
moving with its velocity of propagation, C. In the case of a crystal, the 
upper limit of dislocation velocity is not necessarily C/cose, which 
increases without limit as e-.... TT /2. The approach we employed here is 
simply an analysis of relative motion and does not include the limitations 
-10-
imposed upon dislocation velocities in solids. These, in turn, 
impose an upper limit in time for the integrability of Eq. 19. 
By combining Eqs. 21 and 22, a simple expression for the distance, 
dF, travelled by the dislocation in order to reach a certain fraction of 
the wave velocity is found. If we let v/C = F, then 
2 
d - BC [log ( 1 ) Fcose J . F - bk 2 8 e \ 1 - Fcos9 -a cos (23) 
To conclude this section we consider a dislocation initially moving 
with velocity ~m in the positive~· -direction and interacting with an 
unloading wave moving in the negative ~-direction. In this case, Eq. 16 
reduces to 
= -kdt 
and integrating, 
and 
BC 
T = 
abcose 
{ ( 1 + vmccos9 ) v .. C [ _ abkcose t] _ 1} cos9 exp BC • 
d = BC 2 
abkcos e 
v cose 
m ) ( 1 [ abkcose t) __ c__ t C - exp - B C cos e 
By combining Eqs. 26 and 27, we arrive at an expression for the 
(24) 
(25) 
(26) 
(27) 
distance necessary to bring a dislocation to rest from a fraction F of 
the shear wave velocity. It is 
-11-
BC
2 
[ ] 
---:::;:-2 - Fcose - loge(l + Fcose) _ 
abkcos e 
(28) 
IV. DISCUSSION 
We have treated the most simple case of an advancing plane stress 
wavefront on dislocation motion, that in which ~; lx is constant and 
dislocation velocity is proportional to applied shear stress. Any general 
form for.£!. l or stress-dependence of velocity f[ TR} can be handled, at x 
although closed-form solutions may not be obtainable in all cases. The 
example above, however, is sufficient to point out the importance of 
considering relative motion when v = f[TR} is to be determined 
experimentally. 
As a numerical illustration, we computed the distance required to 
accelerate a dislocation to some fraction F of the wave velocity and 
then bring it to rest, by applying both the approach outlined above, using 
Eqs. 23 and 28, and the classical method where relative motion is ignored 
BC2 2 
and the distance is simply abk F The two cases are compared in Fig. 2 
for e = 0. (In the limit .e = TT /2, the two converge. ) The distance 
computed from the first approach begins to diverge noticeably from that 
of the second when F reaches about 0. 5. The former case gives a result 
similar to that when relative motion is ignored but the velocity-stress 
relation is assumed to be non-linear with the form 
Q''f b , (29) 
where n is a constant greater than zero. (The case n = ~ is plotted in 
Fig. 2.) Therefore if relative velocity between the dislocation and 
-12-
applied stress wave is neglected in the analysis of experimental data, 
then a non-linear damping coefficient will be deduced that is in error 
by a "relativistic•• factor of the form [ 1 - (~ ) 2 Jn. 
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REFERENCES 
1. T. Vreeland, Jr. and K. M. Jassby, Mat. Sci. Engr., ']_, 95 (1971}. 
2. K. M. Jassby and T. Vreeland, Jr., "Direct Measurement of 
Phonon and Electron Damping of Edge Dislocation Motion in Pure 
0 Copper at 4. 2 K 11 , U. S. Atomic· Energy Commission Report 
CALT-767 -P3 -22. 
3. W. G. Johnson and J. J. Gilman, _!G Appl. Phys., ·lQ_, 129 (1959}. 
4. A. P. L. Turner and T. Vreeland, Jr., Acta Met., ~. 1225 (1970}. 
5. J. Weertman and J. Kraft, Dislocation Dynamics, McGraw Hill, 
New York, (1968}, p. 609. 
6. J. Weertman, Response of Metals to High Velocity Deformation, 
Inter science, (1961 }. 
7. J. D. Eshelby, Proc. Phys. Soc., .§2_, 1013 (1956}. 
8. F. R. N. Nabarro, Theory..£% Crystal Dislocations, Oxford Univ. 
Press, (1967} p. 482. 
9. J. P. Hirth and J. Lathe, Theory..£% Dislocations, McGraw Hill, 
New York (1968}, p. 156. 
REVISED FIGURE CAPTION 
Figure 1. Dislocation gliding in the slip plane defined by the normal 
vector ~' x Y..• The slip direction~' is at an angle 6 from 
the direction of propagation of the stress wave. 


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On the measurement of dislocation damping forces at high dislocation velocity

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