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Dislocations moving uniformly on the interface between isotropic media of different elastic properties
J. Mech. Phys. Solids, 1963, Vol. 11, pp. 197 to 204. PergamonPress Ltd. Printedin Great Britain.
DIST,OCATIONS INTERFACE
MOVING RETWEEN
DIFFERENT
ISOTROPIC
ELASTIC Ry
l)cpartmrnt
UNIFORMT,Y
of Matcrinls
(Rccrivrd
J.
THE
MEDIA
OF
PROPERTIES*
WIXKTMAK Sort,hwcst,ern
Scicncc,
20th
ON
Xovrmber,
IYnirersity,
Evanston
1962)
TIIE MASTIC displaccmrnts solution is obtained for SCTCW and for edge dislocations moving on the intcrfacc brtween two isotropic media of different elastic propertics and different, dcnsitics. This problem has application to any diffusionless transformation involving dislocations on the transformation interface. The self-energy of a dislocation is found to become infinite at the slowest sound velocity. The threshold velocity is found to lie between the Rayleigh wave velocities of the two media. This velocity can exist only within a limited range of the elastic constants. The existence of the Kshelby singular dislocation velocity likewise is limited to a range of values of thr elastic con&a&s.
1. ‘rrre
PROBLEM
different
of a dislocation
elastic
practical
properties
viewpoint.
dislocations
I?;TR~DI*CTI~N
moving
on the interface
is interesting
Diffusionless
both
transformations
running on t,he interfaces Since the amount of energy
separating
two media
from the theoretical
between released
in crystals
of
as well as the
probably
involve
transformed and untransformed in such transformations may be
material. large, high dislocation velocities are to be expected. leas proposed that dislocations may run at supersonic
In fact, ~C;SIIIZI.RY (19%) velocities in diffusionlcss
t,rxnsformations. A dislocation
running
is moving
which
separates
two
seems
worthwhile
on a transformation
materials
to obtain
the
of differing solution
interface elastic
of the
properties
stress
on an interface
and
field about
densities. such
It
moving
dislocations. In this paper we shall attempt to solve the problem for the simplest (The assumption of case : that in which the t,wo elastic media are isotropic. isotropy precludes a treatment of the twinning dislocation : in isotropic materials a twinning may have
dislocation is merely an ordinary dislocation. However, the analysis some qualitative application to twinning dislocations in anisotropic
material). The solution of the problem we are considering can be obtained by extending t,hr known solutions of dislocations moving in isotropic material (ESWXI~Y 194Q; FIIAKK 1940; I,EIRFRIED and DIETZ 1949). Consider a co-ordinate system in which
.J. 'V1'lClWLTMAN
198
a dislocation line lies parallel to the z-axis and moves on its slip plane in z-direction. Let pl, A, and pr represent the Lame constants and the density of material above the slip plane (u > 0), and p2, h, and pz the same constants the material below the slip plane (y < 0). For a moving screw dislocation following equations of dynamic equilibrium have to he satisfied :
the the for thr
(1) where i = 1 or 2, 7~:~and ZE’% arc the elastic ~lisplace~neilts in the x-direction above and below the slip plane, es = (&pa)” is the t,ransvrrs:c sot~nd velocity in each of the two media, and t stands for time. For a moving edge dislocation the equations of dynamic equilibrillm arc :
where u.1and zji are the elastic displacements in the x- and y-directions respectively. The time differential 62/S can be replaced by Jr2 a2/&xz for dislocations moving with a uniform velocity Y in the z-direction. From the results of analyses of dislocations moving in isotropic material WC anticipate that the solutions of (I ) and (2) f or moving dislocations will he (a)
Screw dislocation 'it'{ E &
(h)
Edge
Gi tan-' i
dislocation
Ai tan-1 I’$ +R:
!k? '-i_ F:i. 3,' 1
Pi V + Fi, Bitan-l ---+ ,r
(3)
(4n.)
In these equations b is the length of the Burgers vector; Aa, I&, etc. are constants; /3i = (1 - P/c~~)*, yb = (1 - Yz/cAg2) where CAi=- [(At + +)/pr]+ = the longitudinal sound velocity, and z’ = z - Bt where F is the velocity of the dislocations. The constant Hi is added merely to make (Bb) dimensionally correct. The constants & and Fi are added in order to match suitably the elastic displacements 206and U( across the interface ,2/= O. Since only the di~erentials of the displacements are important the constants Es, Ff and IIf can be ignored. The evaluation of the constants for the screw and for the edge dislocations a.re considered separately in the following Sections.
2.
%~REW
~~ISL~~~TIONS
The constants Gs of (3) are simple to evaluate. If a complete circuit is made around a dislocation the elastic displacement must change by an amolmt equal to the Burgers vector. Thus we have from the properties of the arc tan function that
Dislocationsmoving uniformlyon the interfacebetweendiffrrent elastic media G, + G, = 2
199
(5)
when the dislocation velocity Y is less.’ than either cl or c2. Another equation in G1 and G, can be obtained from the condition that the value of the stress must be continuous across the slip plane. Thus at 2 = 0 the stress cVz must satisfy t,he c~ondition (oUz)r - (o~~)~, NOW azei G Bt a’ Pa b (o&l’ = pi = + 2rr x12-j- 82 fp %I
Thercforc
CL< Gz & = t+rG, The
(6)
P,.
fV
only other stress which exists around the screw, namely ozz, is given by
and is equal to zero on the slip plane regardless of the From a soWion of (5) and (‘7) we obtain
values
of G, and G,.
(9n)
(Qbf Equations (9) and (3) give the solution of the elastic displacement field about the moving screw dislocations. From this field both the stresses and the displacement velocities can be found. Once these quantities are known the strain energy and the kinetic energy can be calculated in the usual manner (ESHELBY 1949; FRAKK 1949; LEIBFRIED and DIETZ 1949; WKERTMAN 1961). This ealctdation gives for the total self-energy E of a screw dislocation moving on the interface between two media., each of width K in the ?/-direction and extending to -&:co in the x-direction,
As would be expected, this expression becomes infinite at the slowest of the two t,ransverse sound velocities (i.e. when 8, or ,62 is equal to zero).
3.
EDGE DTSLOCATIOX
The probfem of an edge dislocation moving on an interface separating two different elastic media is more complicated but more interesting than that of the moving screw dislocation. The constants appearing in (4) for the elastic displacements are simply evaluated.
The condition equation
that
the
If (4) is placed into (2) one finds that c, = yg 81,
Wa)
I& = &-’ B;.
Plh)
displacements
describe
an
edge
dislocation
gives
the
200
.J. W~.~~vr.v.~s _A, + 11, + A, -c II, = 2.
The condition
that no line forces act at the dislocation
The reqlCrement the ccluation
that the strcsscs
/L1(yl .I, -I- PI-12 /3-l Finally,
(l%:r)
the corldition
bc continilolls
I&) -
core resldts in
across
thr slip planr
/L2(Yr _I, + X22&’
l)r(!(l~1(~(3
I$) = 0.
t,hat, the plastic tlisplnccmrr~ts arc continllous
(l*e)
a.erosh tlrr slip
plane gives y1_1, i- /3-l
II, -
yz A1, -
/3-l 13, 7 0.
(l*?(l)
Wheu (12) is sol\-ed for _li, etc. one obtains 2
-- p1 $8,x,2)].
(13:1)
where A = & +
(PI
-
PJ
+
81
[Yl
-t
(~2
~2’
-
P~‘J.~‘Y
P2
Y2)
(PI
81
CL2 (PI
+
P2)
(rl
y22
Yl
‘AZ4
+
B2
P22
In order to obtain
Y2
i+ PI2
-
~1~2
CL2 P2) Y2
%‘)
~
(~1
Yl
Y2
+
~2)
/%
P2
(&x22
(PI
+
-
-t- x12 %i2CL1 CL2
(8,
P2
‘x,2)
tL2F Y2
-I-
P2
h)
%“I.
A, and ZZ, one merely
intcrchnngcs
the slthscripts
1 ant1 2
in these cquations.
With these \.alurs of thr constants
Ai, etc. one obtains
the following
expression
for t,he shear stress ozy acting on the slip plane :
It has been found from a study of edge dislocations moving in an isotropic material (WEEHTMAN 1961) that as the dislocation \relocity is increased the shear stress on the slip plane decreases until it becomes zero at the Rayleigh wax-e velocity. At greater velocities it. increases with increasing velocity but has a negative \-nl~lr. For the type of dislocation under consideration the \-elocit)- at which the sheal stress on the slip plane goes to zero can be found by setting (II) equal t,o zero. The following
equation
is obtained
:
Dislocations
moving
uniformly
on the int,erface between
201 (15)
(Yz 82 - CQ")= 0.
(YlPI - XI*)+ gz
2,
different elastic media
The velocity which satisfies this equation is the threshold velocity [we adopted ‘l’~~r~~ro~,-rco’s(IWX) term for this velocity] separating the region of normal dislocation hehaviour (in which dislocations of like sign on the same slip plane repel one another)
from the region of anomalous
behaviour
(dislocations
of like sign attract
one another). The equation yi
pi
-
xi*
=
0
(16)
determines the Rayleigh surface wa\-r velocity in the isot,ropic medium above or below the slip planes. Hence one can see from (15) thnt the threshold velocit? lies between
the Rayleigh
wave velocities
of each medium.
defined by (1.5) is r~ol equal to the Stoneley might
have
Rayleigh
assumed
from
wave velocity
the fact
that
The threshold
wave velocity*, the threshold
when the elastic properties
contrary
velocity
velocity
to what one
is equal
to the
and densities of t,he two media
are identical.
FIG. 1. Plot of the threshold velocity cl v. (pJp,)g. The velocity is expressed in units of cl. Also shown is the variation of the transverse sound velocities cl and c2 and the two Rayleigl~ surface wave velocities crl and erg.
It is not always possible to find a velocity this situation
which satisfies (15). Fig. 1 illustrates
for the simple ease in which the longitudinal
so~md
velocity
in each
medium is very much larger than both c1 and c2 so that both y1 and yz are equal to 1 for velocities near c1 and cz. The densities in the two media are assumed to be equal. One can see from the figure that the occurrence of a threshold velocity is limited to a narrow range in the variables pL1and p2 [(,+_,/~~)a varying from z to 1.1931. If &/CL1is outside this range the anomalous velocity region cannot exist. *Stoneleywnre~(EWIXG,
JARDWTIKY
and
PRXSS
195i)
are the
surfz~~
waves
which
propaaate
along
an interface
202
.J.
method moving
Tl)c
of
the
\%‘kERTMAN
of ~al~~~latior~ used previously can be applied t,o find the energy edge disloc~ation under consideration (WPXRTMAS i 961). OilC~
finds for the strain cnrrgy
fi& the rsprrsaion
and for the kinet,ic energy
RI, the expression
These
expressions reduce to those obtained previously* when pl, pr and A, arc set equal to pz, /ALzand A,. .1s it would be expected, the energies @-en by (17) and (1 S) become infinite at thr slower of the two transverse sound 1.clocitics.
f*, and cZ.
E~xr~~x,nu
(1949) showed that an edge dislocation in an isotropic medimn can 2/ (2) c without a.ny shock wa1.e appearing in its displacement
mo\-c at a velocity field.
This velocity is in a supersonic range and yet the dislocation will not. radiate The reason whv the dislocation can exhibit subsonic behafiour at, this
energy.
supersonic velocity can be seen from (12) when p1 == pZ, A, = ;I,, and p1 = pZ. In this situation (12a) and (12d) are identically zero since A, = A, and R, = 11,. Now R, and R, of (12a) and (12b) must be set equal to zero [and thus eliminate the infinite terms of (JI)] when e < I’ < cA. Equations (12a) and (X2b) may still be satisfied, however, provided the velocity is such that ‘x1 = c(~ = 0. This veIoci$ is y’(2) c. At this velocity no shock ware discontinLlities appear in the stress and displacement field and all of equations (12) are satisfied. We wish to see now what happens to Eshelby’s singular dislocation In order for this singular for the edge dislocation under consideration,
velocit) velocit)
to exist it must be possible to set any of the At and I& of (12) equal to zero when <-he terms n~~~ftiplied by Ai and Bg of (4) contain infinities. If the dislocation velocity lies between c1 and c,, and I3, is t,herefore set equal to zero, (12) cannot. in general hold for any value of T; since we have four equations in three unknown qltantitirs. IIowe\+er, it is possible for all equations to hold if the determinant of the coefficients of (lab) through (12d) is zero. Thus we find the following cquaticn for Eshclhy’s singular dislocation velocity :
I&locations
moving
umfarmly
on the interface
between
different elastic modisl
203
When
p3. = p2 and cAt is very much larger than both c1 and c2 so that -2%-= ‘y2 = X for the velocities ot’ interest, it is found that as &pX is varied from 1 to infmity the E&d@ >rbr v&&y es varies as simwn in Fig. Cr. The sirtguiar vefocity can exist when pA = F~. When p3 is slightly I~~l’ger than pr no singular
4 4 a singular v&city occ\lrs. In the range 2 < pn/px w value is alrtlost equal to the velocity c,,
\-clocity
does exist.
Tts
The behaviour
elastic
media
of dislocations moving on the interface between two &&rent is generally found to be whrtt one would expect from the studies of
d~s~oca~io~~smoving in hom~~enous isotropic mate&& The slowest sound velociky sets an upper limit (apart from the E&&y singular ve1ocit.y-y) to the speed of tltc dislocation since, according to linear elaxtieity theory, the energy became8 infinite at this point. [If it is aswmed that a crystalline material can support only a finite stress it can be shown that supersonic solutions exist. %HELBV (1956)
.J. \Vr.:~,:rc~ntnu
204
and STJIOII (1962) as well as ‘~‘IIO~ISOS (Ig61) ha\ e pitttcd out this fact. damping caused I>\* thr getteratiorr of sho& waves the supersonic 1.elocities \-elocity]. Its
probably are litnited to I-rIocities not rnrtcl~ beyond ‘IVie bkllrlhy singtthr dislocation \-&city tnay
rxistencc
dcpcnds
each medium, at the ICshelby transformation
on
Ikcausr singular
tli?
\~nlucs
energy. vchit!..
disloc~ations
of the
dissipation
rtttt
elastic tlirottglt
cotistants shoeli
Hecattsc 01 dislocation
2/ 2 times a sot~tld or may not exist. and
tltc
wa1.e generat
derisity
this velocity c~~tlrl h that At which \-cr~. fast 011 diffttsiottlcss transformation itttcrfaccs. It
would he ititercstittg. therefore. to see if fast diflrtsionless tratlsformatiotts place itt materials whose c~lastic c~otistattts do trot prtnit the existcticc lSshclh?- singular \.clocity. The tl~resliold \-&city separating a normal from an atiom;llous velocity also may it) the
or may
not csist.
tJv0 isotropic
~S.SII~:IJW, .J. I).
tncclia
It will csist ah\-<’
in
ion is zero
ottly if the two tr:ms\-erse
arid blow
1949 1 !I.50
1 !)3i 1n-L9 1949 1962 1MU’, 1Nil 1 CtOl
tltc slip platir
lit close
sourid
takch of :ttt rcgiotr vcloritics
to each
otlter.
DIST,OCATIONS INTERFACE
MOVING RETWEEN
DIFFERENT
ISOTROPIC
ELASTIC Ry
l)cpartmrnt
UNIFORMT,Y
of Matcrinls
(Rccrivrd
J.
THE
MEDIA
OF
PROPERTIES*
WIXKTMAK Sort,hwcst,ern
Scicncc,
20th
ON
Xovrmber,
IYnirersity,
Evanston
1962)
TIIE MASTIC displaccmrnts solution is obtained for SCTCW and for edge dislocations moving on the intcrfacc brtween two isotropic media of different elastic propertics and different, dcnsitics. This problem has application to any diffusionless transformation involving dislocations on the transformation interface. The self-energy of a dislocation is found to become infinite at the slowest sound velocity. The threshold velocity is found to lie between the Rayleigh wave velocities of the two media. This velocity can exist only within a limited range of the elastic constants. The existence of the Kshelby singular dislocation velocity likewise is limited to a range of values of thr elastic con&a&s.
1. ‘rrre
PROBLEM
different
of a dislocation
elastic
practical
properties
viewpoint.
dislocations
I?;TR~DI*CTI~N
moving
on the interface
is interesting
Diffusionless
both
transformations
running on t,he interfaces Since the amount of energy
separating
two media
from the theoretical
between released
in crystals
of
as well as the
probably
involve
transformed and untransformed in such transformations may be
material. large, high dislocation velocities are to be expected. leas proposed that dislocations may run at supersonic
In fact, ~C;SIIIZI.RY (19%) velocities in diffusionlcss
t,rxnsformations. A dislocation
running
is moving
which
separates
two
seems
worthwhile
on a transformation
materials
to obtain
the
of differing solution
interface elastic
of the
properties
stress
on an interface
and
field about
densities. such
It
moving
dislocations. In this paper we shall attempt to solve the problem for the simplest (The assumption of case : that in which the t,wo elastic media are isotropic. isotropy precludes a treatment of the twinning dislocation : in isotropic materials a twinning may have
dislocation is merely an ordinary dislocation. However, the analysis some qualitative application to twinning dislocations in anisotropic
material). The solution of the problem we are considering can be obtained by extending t,hr known solutions of dislocations moving in isotropic material (ESWXI~Y 194Q; FIIAKK 1940; I,EIRFRIED and DIETZ 1949). Consider a co-ordinate system in which
.J. 'V1'lClWLTMAN
198
a dislocation line lies parallel to the z-axis and moves on its slip plane in z-direction. Let pl, A, and pr represent the Lame constants and the density of material above the slip plane (u > 0), and p2, h, and pz the same constants the material below the slip plane (y < 0). For a moving screw dislocation following equations of dynamic equilibrium have to he satisfied :
the the for thr
(1) where i = 1 or 2, 7~:~and ZE’% arc the elastic ~lisplace~neilts in the x-direction above and below the slip plane, es = (&pa)” is the t,ransvrrs:c sot~nd velocity in each of the two media, and t stands for time. For a moving edge dislocation the equations of dynamic equilibrillm arc :
where u.1and zji are the elastic displacements in the x- and y-directions respectively. The time differential 62/S can be replaced by Jr2 a2/&xz for dislocations moving with a uniform velocity Y in the z-direction. From the results of analyses of dislocations moving in isotropic material WC anticipate that the solutions of (I ) and (2) f or moving dislocations will he (a)
Screw dislocation 'it'{ E &
(h)
Edge
Gi tan-' i
dislocation
Ai tan-1 I’$ +R:
!k? '-i_ F:i. 3,' 1
Pi V + Fi, Bitan-l ---+ ,r
(3)
(4n.)
In these equations b is the length of the Burgers vector; Aa, I&, etc. are constants; /3i = (1 - P/c~~)*, yb = (1 - Yz/cAg2) where CAi=- [(At + +)/pr]+ = the longitudinal sound velocity, and z’ = z - Bt where F is the velocity of the dislocations. The constant Hi is added merely to make (Bb) dimensionally correct. The constants & and Fi are added in order to match suitably the elastic displacements 206and U( across the interface ,2/= O. Since only the di~erentials of the displacements are important the constants Es, Ff and IIf can be ignored. The evaluation of the constants for the screw and for the edge dislocations a.re considered separately in the following Sections.
2.
%~REW
~~ISL~~~TIONS
The constants Gs of (3) are simple to evaluate. If a complete circuit is made around a dislocation the elastic displacement must change by an amolmt equal to the Burgers vector. Thus we have from the properties of the arc tan function that
Dislocationsmoving uniformlyon the interfacebetweendiffrrent elastic media G, + G, = 2
199
(5)
when the dislocation velocity Y is less.’ than either cl or c2. Another equation in G1 and G, can be obtained from the condition that the value of the stress must be continuous across the slip plane. Thus at 2 = 0 the stress cVz must satisfy t,he c~ondition (oUz)r - (o~~)~, NOW azei G Bt a’ Pa b (o&l’ = pi = + 2rr x12-j- 82 fp %I
Thercforc
CL< Gz & = t+rG, The
(6)
P,.
fV
only other stress which exists around the screw, namely ozz, is given by
and is equal to zero on the slip plane regardless of the From a soWion of (5) and (‘7) we obtain
values
of G, and G,.
(9n)
(Qbf Equations (9) and (3) give the solution of the elastic displacement field about the moving screw dislocations. From this field both the stresses and the displacement velocities can be found. Once these quantities are known the strain energy and the kinetic energy can be calculated in the usual manner (ESHELBY 1949; FRAKK 1949; LEIBFRIED and DIETZ 1949; WKERTMAN 1961). This ealctdation gives for the total self-energy E of a screw dislocation moving on the interface between two media., each of width K in the ?/-direction and extending to -&:co in the x-direction,
As would be expected, this expression becomes infinite at the slowest of the two t,ransverse sound velocities (i.e. when 8, or ,62 is equal to zero).
3.
EDGE DTSLOCATIOX
The probfem of an edge dislocation moving on an interface separating two different elastic media is more complicated but more interesting than that of the moving screw dislocation. The constants appearing in (4) for the elastic displacements are simply evaluated.
The condition equation
that
the
If (4) is placed into (2) one finds that c, = yg 81,
Wa)
I& = &-’ B;.
Plh)
displacements
describe
an
edge
dislocation
gives
the
200
.J. W~.~~vr.v.~s _A, + 11, + A, -c II, = 2.
The condition
that no line forces act at the dislocation
The reqlCrement the ccluation
that the strcsscs
/L1(yl .I, -I- PI-12 /3-l Finally,
(l%:r)
the corldition
bc continilolls
I&) -
core resldts in
across
thr slip planr
/L2(Yr _I, + X22&’
l)r(!(l~1(~(3
I$) = 0.
t,hat, the plastic tlisplnccmrr~ts arc continllous
(l*e)
a.erosh tlrr slip
plane gives y1_1, i- /3-l
II, -
yz A1, -
/3-l 13, 7 0.
(l*?(l)
Wheu (12) is sol\-ed for _li, etc. one obtains 2
-- p1 $8,x,2)].
(13:1)
where A = & +
(PI
-
PJ
+
81
[Yl
-t
(~2
~2’
-
P~‘J.~‘Y
P2
Y2)
(PI
81
CL2 (PI
+
P2)
(rl
y22
Yl
‘AZ4
+
B2
P22
In order to obtain
Y2
i+ PI2
-
~1~2
CL2 P2) Y2
%‘)
~
(~1
Yl
Y2
+
~2)
/%
P2
(&x22
(PI
+
-
-t- x12 %i2CL1 CL2
(8,
P2
‘x,2)
tL2F Y2
-I-
P2
h)
%“I.
A, and ZZ, one merely
intcrchnngcs
the slthscripts
1 ant1 2
in these cquations.
With these \.alurs of thr constants
Ai, etc. one obtains
the following
expression
for t,he shear stress ozy acting on the slip plane :
It has been found from a study of edge dislocations moving in an isotropic material (WEEHTMAN 1961) that as the dislocation \relocity is increased the shear stress on the slip plane decreases until it becomes zero at the Rayleigh wax-e velocity. At greater velocities it. increases with increasing velocity but has a negative \-nl~lr. For the type of dislocation under consideration the \-elocit)- at which the sheal stress on the slip plane goes to zero can be found by setting (II) equal t,o zero. The following
equation
is obtained
:
Dislocations
moving
uniformly
on the int,erface between
201 (15)
(Yz 82 - CQ")= 0.
(YlPI - XI*)+ gz
2,
different elastic media
The velocity which satisfies this equation is the threshold velocity [we adopted ‘l’~~r~~ro~,-rco’s(IWX) term for this velocity] separating the region of normal dislocation hehaviour (in which dislocations of like sign on the same slip plane repel one another)
from the region of anomalous
behaviour
(dislocations
of like sign attract
one another). The equation yi
pi
-
xi*
=
0
(16)
determines the Rayleigh surface wa\-r velocity in the isot,ropic medium above or below the slip planes. Hence one can see from (15) thnt the threshold velocit? lies between
the Rayleigh
wave velocities
of each medium.
defined by (1.5) is r~ol equal to the Stoneley might
have
Rayleigh
assumed
from
wave velocity
the fact
that
The threshold
wave velocity*, the threshold
when the elastic properties
contrary
velocity
velocity
to what one
is equal
to the
and densities of t,he two media
are identical.
FIG. 1. Plot of the threshold velocity cl v. (pJp,)g. The velocity is expressed in units of cl. Also shown is the variation of the transverse sound velocities cl and c2 and the two Rayleigl~ surface wave velocities crl and erg.
It is not always possible to find a velocity this situation
which satisfies (15). Fig. 1 illustrates
for the simple ease in which the longitudinal
so~md
velocity
in each
medium is very much larger than both c1 and c2 so that both y1 and yz are equal to 1 for velocities near c1 and cz. The densities in the two media are assumed to be equal. One can see from the figure that the occurrence of a threshold velocity is limited to a narrow range in the variables pL1and p2 [(,+_,/~~)a varying from z to 1.1931. If &/CL1is outside this range the anomalous velocity region cannot exist. *Stoneleywnre~(EWIXG,
JARDWTIKY
and
PRXSS
195i)
are the
surfz~~
waves
which
propaaate
along
an interface
202
.J.
method moving
Tl)c
of
the
\%‘kERTMAN
of ~al~~~latior~ used previously can be applied t,o find the energy edge disloc~ation under consideration (WPXRTMAS i 961). OilC~
finds for the strain cnrrgy
fi& the rsprrsaion
and for the kinet,ic energy
RI, the expression
These
expressions reduce to those obtained previously* when pl, pr and A, arc set equal to pz, /ALzand A,. .1s it would be expected, the energies @-en by (17) and (1 S) become infinite at thr slower of the two transverse sound 1.clocitics.
f*, and cZ.
E~xr~~x,nu
(1949) showed that an edge dislocation in an isotropic medimn can 2/ (2) c without a.ny shock wa1.e appearing in its displacement
mo\-c at a velocity field.
This velocity is in a supersonic range and yet the dislocation will not. radiate The reason whv the dislocation can exhibit subsonic behafiour at, this
energy.
supersonic velocity can be seen from (12) when p1 == pZ, A, = ;I,, and p1 = pZ. In this situation (12a) and (12d) are identically zero since A, = A, and R, = 11,. Now R, and R, of (12a) and (12b) must be set equal to zero [and thus eliminate the infinite terms of (JI)] when e < I’ < cA. Equations (12a) and (X2b) may still be satisfied, however, provided the velocity is such that ‘x1 = c(~ = 0. This veIoci$ is y’(2) c. At this velocity no shock ware discontinLlities appear in the stress and displacement field and all of equations (12) are satisfied. We wish to see now what happens to Eshelby’s singular dislocation In order for this singular for the edge dislocation under consideration,
velocit) velocit)
to exist it must be possible to set any of the At and I& of (12) equal to zero when <-he terms n~~~ftiplied by Ai and Bg of (4) contain infinities. If the dislocation velocity lies between c1 and c,, and I3, is t,herefore set equal to zero, (12) cannot. in general hold for any value of T; since we have four equations in three unknown qltantitirs. IIowe\+er, it is possible for all equations to hold if the determinant of the coefficients of (lab) through (12d) is zero. Thus we find the following cquaticn for Eshclhy’s singular dislocation velocity :
I&locations
moving
umfarmly
on the interface
between
different elastic modisl
203
When
p3. = p2 and cAt is very much larger than both c1 and c2 so that -2%-= ‘y2 = X for the velocities ot’ interest, it is found that as &pX is varied from 1 to infmity the E&d@ >rbr v&&y es varies as simwn in Fig. Cr. The sirtguiar vefocity can exist when pA = F~. When p3 is slightly I~~l’ger than pr no singular
4 4 a singular v&city occ\lrs. In the range 2 < pn/px w value is alrtlost equal to the velocity c,,
\-clocity
does exist.
Tts
The behaviour
elastic
media
of dislocations moving on the interface between two &&rent is generally found to be whrtt one would expect from the studies of
d~s~oca~io~~smoving in hom~~enous isotropic mate&& The slowest sound velociky sets an upper limit (apart from the E&&y singular ve1ocit.y-y) to the speed of tltc dislocation since, according to linear elaxtieity theory, the energy became8 infinite at this point. [If it is aswmed that a crystalline material can support only a finite stress it can be shown that supersonic solutions exist. %HELBV (1956)
.J. \Vr.:~,:rc~ntnu
204
and STJIOII (1962) as well as ‘~‘IIO~ISOS (Ig61) ha\ e pitttcd out this fact. damping caused I>\* thr getteratiorr of sho& waves the supersonic 1.elocities \-elocity]. Its
probably are litnited to I-rIocities not rnrtcl~ beyond ‘IVie bkllrlhy singtthr dislocation \-&city tnay
rxistencc
dcpcnds
each medium, at the ICshelby transformation
on
Ikcausr singular
tli?
\~nlucs
energy. vchit!..
disloc~ations
of the
dissipation
rtttt
elastic tlirottglt
cotistants shoeli
Hecattsc 01 dislocation
2/ 2 times a sot~tld or may not exist. and
tltc
wa1.e generat
derisity
this velocity c~~tlrl h that At which \-cr~. fast 011 diffttsiottlcss transformation itttcrfaccs. It
would he ititercstittg. therefore. to see if fast diflrtsionless tratlsformatiotts place itt materials whose c~lastic c~otistattts do trot prtnit the existcticc lSshclh?- singular \.clocity. The tl~resliold \-&city separating a normal from an atiom;llous velocity also may it) the
or may
not csist.
tJv0 isotropic
~S.SII~:IJW, .J. I).
tncclia
It will csist ah\-<’
in
ion is zero
ottly if the two tr:ms\-erse
arid blow
1949 1 !I.50
1 !)3i 1n-L9 1949 1962 1MU’, 1Nil 1 CtOl
tltc slip platir
lit close
sourid
takch of :ttt rcgiotr vcloritics
to each
otlter.