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The rate-determining process in Luders band growth
Scripta
METALLURGICA
Vol. Printed
6, pp. 677-680, in the United
1972 States
Pergamon
Press,
Inc.
THE RATE-DETERMINING PROCESS IN LUDERS BAND GROWTH
A.W. Sleeswyk and D.J.Verel Laboratorium voor Fysische Metaalkunde, Materials Science Centre, Universiteitscomplex Paddepoel, Groningen, The Netherlands.
(Received
June
9,
1972)
What process determines the rate of increase of a steadily growing Luders band? Hahn (]), who investigated the problem
a decade ago, believed that it was the injection of mobile dis-
locations from the moving Luders band front into the region ahead of it. According to his model the velocity, u, of the band front is determined by the speed v. with which the 3 injected dislocations move, i.e.: u ~ k.v. J
(])
where k.v. is the component of v. normal to the band front. 3 J The results of Sleeswyk's more recent Arrhenius-type analysis
(2) of experimental data
on the upper yield point OUy and the lower yield stress oLy would indicate that Hahn's model applies to the ~ 2 ~
of Luders bands, but not to the steady state propagation of
the Luders band front. This analysis is based on the Arrhenius equation for plastic flow:
-H(a)
= ~c exp
kT '
(2)
which can be used to obtain [c-values from experimental data. Using the well-known equation: = ~.0.b.v,
(3)
where: p, b and v are the density, the Burgers vector and the velocity of the moving dislocations respectively,
~
could be interpreted as the value that ~ would have if v would C
equal the shear wave velocity. With this interpretation,
the following dislocation densities
were derived from °UY and oLY measurements on low-carbon Fe-alloys available in the literature OUy measurements by Hendrickson and Wood (3): ¢c = 6 . 3 × I0 ~ s e c - 1 ;
p = 1.7 x lO 7 cm- 2
aLy
. . . .
Lean et al. (4)
: ~c
=
1.0
x
lO 10 "
;
p
=
2.7
x
lO 12 "
aLy
. . . .
Rosenfield and Hahn (5) : ~c
=
2.5
x
lO 9
;
0
=
6.7
x
lO II
"
"
These o-values may be compared with the values of the total dislocation densities of 10 6 to 10 8 cm -2 for annealed low-carbon steel (I), and of 1010 to I0 II cm -2 for the same alloys after passage of a L~ders band (6,7,8). The high dislocation densities are observed
677
678
RATE-DETERMINING
PROCESS
IN L U D E R S
BAND
GROWTH
Vol.
6, No. 8
only behind the L~ders band front, i.e. the injected dislocations ahead of it do not significantly alter the dislocation density of the annealed material in Hahn's model. As the upper yield point phenomenon is exclusively associated with the nucleation of a L~ders band, the result of the Arrhenius-type analysis would indicate that the model of Hahn may represent the rate-determining process for Luders band nucleation, but that for the steady-state propagation of the Luders band front some other process must be rate-determining.
This con-
clusion does not preclude the successful explanation of the delay time for the onset of yielding furnished by Hahn's model. Investigations by Owen et al. (9) and by Taylor and Malvern (10) of the strain distribution in the L~ders band show that behind the band front the plastic strain increases from zero approximately linearly in a region of which the width, r, is 0.5 to 5 mr,. Beyond this region, which will be called the 'active zone' in what follows, the strain rapidly approaches a constant value. It follows from the steady state motion of the band front that the strain rate ~ in the active zone is approximately a constant, and that outside it ~ may be neglected. The strain rate in the active zone is then simply related to the relative velocity V of the heads of the specimen: V =
-. r
(4)
It was shown by Hart (ll) that V is equal to the product of the L~ders strain eL and u: V = u . E L.
(5)
Substituting eq.(5) in eq.(4), = U'eL r
(6)
is obtained. We introduce now a model for the rate-determining process during L~ders band growth, which is based on the following two assumptions: a. all dislocation multiplication in the active zone takes place at the L~ders band front, i.e. at the interface of the active zone and the macroscopically undeformed material. b. the large majority of the newly created dislocations in the active zone have a limited mean free path L. When these dislocations are halted or when their velocity has become negligibly small, they are in the L~ders band at the back of the active zone, i.e. the active zone ends where the dislocations have ceased to move. If the average dislocation velocity in the active zone is v, the average time t that the dislocatlons move along their free paths is: t = L- - . v
(7)
This time may be equated to the time that the dislocations are in the active zone: t
=
r u
-.
(8)
The equations (3), (6), (7) and (8) may be combined, giving an expression for L: 2.e L e =0-'~--"
(9)
Vol.
6, No.
8
RATE-DETERMINING
PROCESS
IN L U D E R S
BAND
GROWTH
679
Substituting in eq.(9) the values: EL % 0.0], p % I0 I0 em -2, b = 2.5 × 10-8 cm, a value of L % l0-4 cm results. This L-value, being a few times the distance between dislocations, appears to be physically plausible. The physical cause of the limitation on the paths of the dislocations suggests itself readily: a fraction f of the moving dislocations that intersect each other become jogged, as a result of which the dislocations move more slowly. After a number i of intersections the dislocations have slowed down to a velocity that is either zero or negligibly small. The rate of decrease in dislocation velocity that is inherent to this model is magnified by the effect of pinning due to the diffusion of impurity atoms. This would explain the abruptness with which the dislocations are halted at the end of their free paths. From this simple model another expression for the dislocation mean free path results: e =
i
(10)
f.pl Estimating: f ~ I, substitution of eq.(lO) in eq.(9) results in:
0 =~x-"~b)2.
(11)
With: i = I, this expression gives an upper limit to the density of mobile dislocations in the active zone. Taking the values of EL and b that were used in assessing the physical plausibility of eq.(9), the value: p % ]0 II cm -2 is obtained, which is, within the limits of accuracy, in accord with the experimentally determined dislocatioh densities, and with the results of the Arrhenius-type analysis of ~LY data. Apparently, it is of common occurence during Luders band propagation that the dislocations are effectively halted at their first intersection.
Acknowledsement This study was performed as part of a project jointly sponsored by the Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.) and the Metaalinstituut T.N.O. at Delft.
References I. G.T.Hahn, Acta Met. I0, 727 (]962). 2. A.W. Sleeswyk, Scripta Met. 4, 355 (]970). 3. J.A.Hendrickson and D.S.Wood, Trans.Amer. Soc.Met. 50, 498 (1958). 4. J.B.Lean, J.Plateau and C.Crussard, Compt.rend.
247, ]458 (]958).
5. A.R. Rosenfield and G.T.Hahn, Trans.Amer. Soc.Met. 59, 962 (1966). 6. A.S.Keh, Phil.Mag. 7. W.F.Flanagan,
12, 9 (]965).
Sc.D.Thesis, MIT (]959).
8. W. Carrington, K.F.Hale and D.MacLean, Proc.Roy. Soc. A259, 203 (1964). 9. W.S.Owen, B.L.Averbach and M.Cohen, Trans.Amer.lnst.of Min.(Metall.)Engrs
50, 634 (1958).
]0. D.B.C.Taylor and L.E.Malvern, 'Response of Metals to High Velocity Deformation', p.77, Interscience Publ., New York (196]). l]. E.W.Hart, Acta Met. 3, 146 (1955).
METALLURGICA
Vol. Printed
6, pp. 677-680, in the United
1972 States
Pergamon
Press,
Inc.
THE RATE-DETERMINING PROCESS IN LUDERS BAND GROWTH
A.W. Sleeswyk and D.J.Verel Laboratorium voor Fysische Metaalkunde, Materials Science Centre, Universiteitscomplex Paddepoel, Groningen, The Netherlands.
(Received
June
9,
1972)
What process determines the rate of increase of a steadily growing Luders band? Hahn (]), who investigated the problem
a decade ago, believed that it was the injection of mobile dis-
locations from the moving Luders band front into the region ahead of it. According to his model the velocity, u, of the band front is determined by the speed v. with which the 3 injected dislocations move, i.e.: u ~ k.v. J
(])
where k.v. is the component of v. normal to the band front. 3 J The results of Sleeswyk's more recent Arrhenius-type analysis
(2) of experimental data
on the upper yield point OUy and the lower yield stress oLy would indicate that Hahn's model applies to the ~ 2 ~
of Luders bands, but not to the steady state propagation of
the Luders band front. This analysis is based on the Arrhenius equation for plastic flow:
-H(a)
= ~c exp
kT '
(2)
which can be used to obtain [c-values from experimental data. Using the well-known equation: = ~.0.b.v,
(3)
where: p, b and v are the density, the Burgers vector and the velocity of the moving dislocations respectively,
~
could be interpreted as the value that ~ would have if v would C
equal the shear wave velocity. With this interpretation,
the following dislocation densities
were derived from °UY and oLY measurements on low-carbon Fe-alloys available in the literature OUy measurements by Hendrickson and Wood (3): ¢c = 6 . 3 × I0 ~ s e c - 1 ;
p = 1.7 x lO 7 cm- 2
aLy
. . . .
Lean et al. (4)
: ~c
=
1.0
x
lO 10 "
;
p
=
2.7
x
lO 12 "
aLy
. . . .
Rosenfield and Hahn (5) : ~c
=
2.5
x
lO 9
;
0
=
6.7
x
lO II
"
"
These o-values may be compared with the values of the total dislocation densities of 10 6 to 10 8 cm -2 for annealed low-carbon steel (I), and of 1010 to I0 II cm -2 for the same alloys after passage of a L~ders band (6,7,8). The high dislocation densities are observed
677
678
RATE-DETERMINING
PROCESS
IN L U D E R S
BAND
GROWTH
Vol.
6, No. 8
only behind the L~ders band front, i.e. the injected dislocations ahead of it do not significantly alter the dislocation density of the annealed material in Hahn's model. As the upper yield point phenomenon is exclusively associated with the nucleation of a L~ders band, the result of the Arrhenius-type analysis would indicate that the model of Hahn may represent the rate-determining process for Luders band nucleation, but that for the steady-state propagation of the Luders band front some other process must be rate-determining.
This con-
clusion does not preclude the successful explanation of the delay time for the onset of yielding furnished by Hahn's model. Investigations by Owen et al. (9) and by Taylor and Malvern (10) of the strain distribution in the L~ders band show that behind the band front the plastic strain increases from zero approximately linearly in a region of which the width, r, is 0.5 to 5 mr,. Beyond this region, which will be called the 'active zone' in what follows, the strain rapidly approaches a constant value. It follows from the steady state motion of the band front that the strain rate ~ in the active zone is approximately a constant, and that outside it ~ may be neglected. The strain rate in the active zone is then simply related to the relative velocity V of the heads of the specimen: V =
-. r
(4)
It was shown by Hart (ll) that V is equal to the product of the L~ders strain eL and u: V = u . E L.
(5)
Substituting eq.(5) in eq.(4), = U'eL r
(6)
is obtained. We introduce now a model for the rate-determining process during L~ders band growth, which is based on the following two assumptions: a. all dislocation multiplication in the active zone takes place at the L~ders band front, i.e. at the interface of the active zone and the macroscopically undeformed material. b. the large majority of the newly created dislocations in the active zone have a limited mean free path L. When these dislocations are halted or when their velocity has become negligibly small, they are in the L~ders band at the back of the active zone, i.e. the active zone ends where the dislocations have ceased to move. If the average dislocation velocity in the active zone is v, the average time t that the dislocatlons move along their free paths is: t = L- - . v
(7)
This time may be equated to the time that the dislocations are in the active zone: t
=
r u
-.
(8)
The equations (3), (6), (7) and (8) may be combined, giving an expression for L: 2.e L e =0-'~--"
(9)
Vol.
6, No.
8
RATE-DETERMINING
PROCESS
IN L U D E R S
BAND
GROWTH
679
Substituting in eq.(9) the values: EL % 0.0], p % I0 I0 em -2, b = 2.5 × 10-8 cm, a value of L % l0-4 cm results. This L-value, being a few times the distance between dislocations, appears to be physically plausible. The physical cause of the limitation on the paths of the dislocations suggests itself readily: a fraction f of the moving dislocations that intersect each other become jogged, as a result of which the dislocations move more slowly. After a number i of intersections the dislocations have slowed down to a velocity that is either zero or negligibly small. The rate of decrease in dislocation velocity that is inherent to this model is magnified by the effect of pinning due to the diffusion of impurity atoms. This would explain the abruptness with which the dislocations are halted at the end of their free paths. From this simple model another expression for the dislocation mean free path results: e =
i
(10)
f.pl Estimating: f ~ I, substitution of eq.(lO) in eq.(9) results in:
0 =~x-"~b)2.
(11)
With: i = I, this expression gives an upper limit to the density of mobile dislocations in the active zone. Taking the values of EL and b that were used in assessing the physical plausibility of eq.(9), the value: p % ]0 II cm -2 is obtained, which is, within the limits of accuracy, in accord with the experimentally determined dislocatioh densities, and with the results of the Arrhenius-type analysis of ~LY data. Apparently, it is of common occurence during Luders band propagation that the dislocations are effectively halted at their first intersection.
Acknowledsement This study was performed as part of a project jointly sponsored by the Stichting voor Fundamenteel Onderzoek der Materie (F.O.M.) and the Metaalinstituut T.N.O. at Delft.
References I. G.T.Hahn, Acta Met. I0, 727 (]962). 2. A.W. Sleeswyk, Scripta Met. 4, 355 (]970). 3. J.A.Hendrickson and D.S.Wood, Trans.Amer. Soc.Met. 50, 498 (1958). 4. J.B.Lean, J.Plateau and C.Crussard, Compt.rend.
247, ]458 (]958).
5. A.R. Rosenfield and G.T.Hahn, Trans.Amer. Soc.Met. 59, 962 (1966). 6. A.S.Keh, Phil.Mag. 7. W.F.Flanagan,
12, 9 (]965).
Sc.D.Thesis, MIT (]959).
8. W. Carrington, K.F.Hale and D.MacLean, Proc.Roy. Soc. A259, 203 (1964). 9. W.S.Owen, B.L.Averbach and M.Cohen, Trans.Amer.lnst.of Min.(Metall.)Engrs
50, 634 (1958).
]0. D.B.C.Taylor and L.E.Malvern, 'Response of Metals to High Velocity Deformation', p.77, Interscience Publ., New York (196]). l]. E.W.Hart, Acta Met. 3, 146 (1955).