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Bemerkung zur Eindeutigkeit der Boltzmannschen Lösung der eindimensionalen Diffusionsgleichung
LETTERS
TO THE
EDITOR
259
TABLE 1
Grain size
q, yield stress (lb/in*)
= Iv,1 total Liiders front speed (mm/set)
sL yield point elongation
10
28,000-28,800 26,300-27,400 25,500-25,800
1.89 2.30 2.99
4.5 3.7 2.8
2 4 10 20
37,800-39,800 36,900-37,100 35,200-35,400 33,900-34,100
1.10 1.33 1.48 1.76
7.7 6.4 5.7 4.8
Number of Liiders fronts
-__
True stress during homogeneous deformation (lb/ins) .__._
0.020 mm
2
0.007 mm
44,700 44,800 44,500 44,400
Strain rate, E = 0.04 min-‘(0.0066 set-I). Gauge length, 1, = 5in.(127 mm).
length. These data are listed in Table 1. The flow stress curves for homogeneous deformation were very similar for samples of comparable grain size even though the yield stress was changed considerably by varying the number of Liiders fronts. This point is illustrated in Table 1 by the similarity of the true stresses on the specimens at two values of uniform strain. The data have their significance in the interpretation of the lower yield stress for the discontinuous yielding of iron and other body-centered cubic metals. Lower yield stresses or the stresses necessary to propagate Liiders bands can be compared only among specimens which show a fixed number of simple Liiders fronts. For example, in plotting the dependence of the lower yield stress on the grain size in the manner of Petchc4), all samples should contain the same number of Liiders fronts. The yield stress for large grain sizes with multiple Liiders fronts can be considerably lower than the stress predicted by the extrapolation of the
yield stress-grain size relationship from fine grain sizes where only two Liiders fronts are p&sent. The slope of the yield stress versus the reciprocal square root of grain size plotted from the data of the present study for specimens with two Liiders fronts is approximately 10 per cent lower than the slope reported by Codd and Petch(5) for a similar steel. A qualitative explanation of the stress drop which results from the nucleation of additional Liiders bands may lie in the increased number of dislocations which can participate in the deformation. With an increased number of active dislocations located at the new Liiders fronts, the same specimen strain rate can be maintained by the motion of dislocations at lower velocity which in turn requires a smaller applied stress.‘@ Graham Research Laboratory & Laughlin Steel Corp.
J. F. BUTLER
Jones
Pittsburgh, Pa.
References 1. W. M. LOMER,J. Me&. Phys. S&da 1, 64 (1952). 2. J. C. FISHERand H. C. ROGERS,Acta Met. 4, 180 (1966). 3. D. S. CLARKand D. S. WOOD, Proc. Amer. Sm. Test. Mat. 49, 717 (1949). 4. N. J. PETCR,J. Iron St. Inat. 173, 25 (1953). 5. I. CODD and N. J. PETCH, Phil Mag. 5, 30 (1960). 6. W. G. JOHNSONand J. J. GILMAN, J. Appl. Phys. 80, 129 (1969). * Received September 29, 1961; revised October 20, 1961.
x
z
30-
b’
25-
GRAIN
SIZE
0.020 mm.
Bemerkung zur Eindeutigkeit der Boltzmannschen Liisung der eindimensionalen Diffusionsgleichung* Fiir die Differentialgleichung der eindimensionelen Diffusion
20
t
151
’ 2
I
I
4 NUMBER
IO OF
LijDERS
I
20 FRONTS
FIG. 2. The dependence of the lower yield stress on the number of Liiders fronts present in the sample. 6
au a Da” -=at
axf ax
1
(1)
mit konzentrationssbh&ngigem Diffusionskoeffizienten
ACTA
280
METALLURQICA,
II(zl) gab Boltzmann(l) eine Losung unter den Anfangsbedingungen u=uO
fur
o
u=z$
fiir
-c0
(2)
VOL.
10,
1962
Literatur 1. L. BOLTZMANN, Ann. Phys., Lpz. 58,959
(1894). 2. J. S. KIRKALDY, Acta Met. 4, 92 (1966). 3. C. SEYEERTH, 2. Angew. Math. Me&. 99, 441 (1959). 4. C. SEYFERTH, Math. Nacho., 24, Heft 1 (1962). * Eingegangen 4. Oktober 1961.
Kirkaldy’s) stellt klar, da3 diese Losung, die nur von x/d/t abhiingt, nicht nur eine Niiherung, wie mitunter
angenommen wurde, sondern eine exakte Losung ist. Orientation relation between austenite and martensite in an Fe-Ni alloy transformed by Dariiber hinaus beheuptet er, sie sei eindeutig, gibt isothermal compressiont aber keinen Beweis an. 1. On cooling nickel-iron with about 30 percent An dem folgenden Beispiel wird nun gezeigt, da13 nickel, the face-centred cubic (f.c.c.) y-phase is die Differentialgleichung (1) unter den Randbedintransformed into the body-centred cubic (b.c.c.) a’gungen (2) mehr sls eine Losung haben kann: phase by a martensitic type of reaction. The orientation relationship usually observed was determined ES sei D = const. Denn kann man fiir die Boltzmannsche Losung first by Nishiyama’n, i.e.
I
uOfUq[l.-erf(&)],fallst>O
U(5, t) =
uo, falls t = 0,
I% falls t =
0,
(3)
2 > 0 x < 0
mit ‘exp (erf y = -? 2/r s 0
q2) dy
(GauBsche Fehlerfunktion) schreiben. Daneben erfiillt aber z.B. such U(z, t) + -$ exp - -& v(x, t) =
(
i ug, falls t = 0,
x > 0
[ 211,falls t = 0,
2 < 0
1
1
, falls t > 0
1
v(O, 4 + =J, wenn t 3
G-4
+O.
Beide Lijsungen (3) und (5) sind fur -co < x < + co und t 2 0 mit Ausnahme von x = 0, t = 0 stetig und besitzen in -co < x < + co, t > 0 alle stetigen partiellen Ableitungen bis zur 2. Ordnung. Die Voraussetzungen, die Kirkaldy macht, sind also fiir die Eindeutigkeit nicht hinreichend. Mit einer Beschr&nktheitsbedingung fiir u wurden inzwischen Eindeutigkeitsbeweise fiir Liisungen der Differentialgleichung (1) unter den Randbedingungen (2)(n und unter allgemeineren Randbedingungen(*) erbracht. C. SEYFERTH Physikalisch- Technischee Irwtitut Berlin
These results were confirmed by Nishiyamac3), who found moreover in similarly thin specimens the orientation relationship originally proposed by Bain, i.e.
t J
die Differentialgleichung (1) und die Randbedingungen (2). Diese Fur&ion hat die pathologische Eigenschaft
der Deutschen Akademie
Recently Pitschc2), working with test-pieces of such thinness that electron-diffraction patterns could be obtained, pointed out that a different orientation relationship was present, i.e.
der Wissenschuften
Nishiyama remarks, however, that the accuracy of his electron-diffraction patterns is not more than 5”. This is, indeed, insufficient to state with certainty whether the Pitsch or the Bsin relation is actually present, for this inaccuracy is of the same order of magnitude as the difference in orientation between the three relations mentioned above. 2. In view of the interesting suggestion made by Bain(*) that the transition of f.c.c. austenite into b.c.c. martensite might be brought about by a simple compression parallel to one of the cube axes of the originally face-centred cubic lattice, we prepared iron-nickel single crystals with a composition of 30.2 wt.% Ni and subjected them to a compressive force at room temperature. The compression experiments were carried out with circular discs with a t This work is part of the program of the Dutch Foundation of Fundamental Research (F.O.M.-T.N.O.), research group Prof. Dr. W. G. Burgers.
TO THE
EDITOR
259
TABLE 1
Grain size
q, yield stress (lb/in*)
= Iv,1 total Liiders front speed (mm/set)
sL yield point elongation
10
28,000-28,800 26,300-27,400 25,500-25,800
1.89 2.30 2.99
4.5 3.7 2.8
2 4 10 20
37,800-39,800 36,900-37,100 35,200-35,400 33,900-34,100
1.10 1.33 1.48 1.76
7.7 6.4 5.7 4.8
Number of Liiders fronts
-__
True stress during homogeneous deformation (lb/ins) .__._
0.020 mm
2
0.007 mm
44,700 44,800 44,500 44,400
Strain rate, E = 0.04 min-‘(0.0066 set-I). Gauge length, 1, = 5in.(127 mm).
length. These data are listed in Table 1. The flow stress curves for homogeneous deformation were very similar for samples of comparable grain size even though the yield stress was changed considerably by varying the number of Liiders fronts. This point is illustrated in Table 1 by the similarity of the true stresses on the specimens at two values of uniform strain. The data have their significance in the interpretation of the lower yield stress for the discontinuous yielding of iron and other body-centered cubic metals. Lower yield stresses or the stresses necessary to propagate Liiders bands can be compared only among specimens which show a fixed number of simple Liiders fronts. For example, in plotting the dependence of the lower yield stress on the grain size in the manner of Petchc4), all samples should contain the same number of Liiders fronts. The yield stress for large grain sizes with multiple Liiders fronts can be considerably lower than the stress predicted by the extrapolation of the
yield stress-grain size relationship from fine grain sizes where only two Liiders fronts are p&sent. The slope of the yield stress versus the reciprocal square root of grain size plotted from the data of the present study for specimens with two Liiders fronts is approximately 10 per cent lower than the slope reported by Codd and Petch(5) for a similar steel. A qualitative explanation of the stress drop which results from the nucleation of additional Liiders bands may lie in the increased number of dislocations which can participate in the deformation. With an increased number of active dislocations located at the new Liiders fronts, the same specimen strain rate can be maintained by the motion of dislocations at lower velocity which in turn requires a smaller applied stress.‘@ Graham Research Laboratory & Laughlin Steel Corp.
J. F. BUTLER
Jones
Pittsburgh, Pa.
References 1. W. M. LOMER,J. Me&. Phys. S&da 1, 64 (1952). 2. J. C. FISHERand H. C. ROGERS,Acta Met. 4, 180 (1966). 3. D. S. CLARKand D. S. WOOD, Proc. Amer. Sm. Test. Mat. 49, 717 (1949). 4. N. J. PETCR,J. Iron St. Inat. 173, 25 (1953). 5. I. CODD and N. J. PETCH, Phil Mag. 5, 30 (1960). 6. W. G. JOHNSONand J. J. GILMAN, J. Appl. Phys. 80, 129 (1969). * Received September 29, 1961; revised October 20, 1961.
x
z
30-
b’
25-
GRAIN
SIZE
0.020 mm.
Bemerkung zur Eindeutigkeit der Boltzmannschen Liisung der eindimensionalen Diffusionsgleichung* Fiir die Differentialgleichung der eindimensionelen Diffusion
20
t
151
’ 2
I
I
4 NUMBER
IO OF
LijDERS
I
20 FRONTS
FIG. 2. The dependence of the lower yield stress on the number of Liiders fronts present in the sample. 6
au a Da” -=at
axf ax
1
(1)
mit konzentrationssbh&ngigem Diffusionskoeffizienten
ACTA
280
METALLURQICA,
II(zl) gab Boltzmann(l) eine Losung unter den Anfangsbedingungen u=uO
fur
o
u=z$
fiir
-c0
(2)
VOL.
10,
1962
Literatur 1. L. BOLTZMANN, Ann. Phys., Lpz. 58,959
(1894). 2. J. S. KIRKALDY, Acta Met. 4, 92 (1966). 3. C. SEYEERTH, 2. Angew. Math. Me&. 99, 441 (1959). 4. C. SEYFERTH, Math. Nacho., 24, Heft 1 (1962). * Eingegangen 4. Oktober 1961.
Kirkaldy’s) stellt klar, da3 diese Losung, die nur von x/d/t abhiingt, nicht nur eine Niiherung, wie mitunter
angenommen wurde, sondern eine exakte Losung ist. Orientation relation between austenite and martensite in an Fe-Ni alloy transformed by Dariiber hinaus beheuptet er, sie sei eindeutig, gibt isothermal compressiont aber keinen Beweis an. 1. On cooling nickel-iron with about 30 percent An dem folgenden Beispiel wird nun gezeigt, da13 nickel, the face-centred cubic (f.c.c.) y-phase is die Differentialgleichung (1) unter den Randbedintransformed into the body-centred cubic (b.c.c.) a’gungen (2) mehr sls eine Losung haben kann: phase by a martensitic type of reaction. The orientation relationship usually observed was determined ES sei D = const. Denn kann man fiir die Boltzmannsche Losung first by Nishiyama’n, i.e.
I
uOfUq[l.-erf(&)],fallst>O
U(5, t) =
uo, falls t = 0,
I% falls t =
0,
(3)
2 > 0 x < 0
mit ‘exp (erf y = -? 2/r s 0
q2) dy
(GauBsche Fehlerfunktion) schreiben. Daneben erfiillt aber z.B. such U(z, t) + -$ exp - -& v(x, t) =
(
i ug, falls t = 0,
x > 0
[ 211,falls t = 0,
2 < 0
1
1
, falls t > 0
1
v(O, 4 + =J, wenn t 3
G-4
+O.
Beide Lijsungen (3) und (5) sind fur -co < x < + co und t 2 0 mit Ausnahme von x = 0, t = 0 stetig und besitzen in -co < x < + co, t > 0 alle stetigen partiellen Ableitungen bis zur 2. Ordnung. Die Voraussetzungen, die Kirkaldy macht, sind also fiir die Eindeutigkeit nicht hinreichend. Mit einer Beschr&nktheitsbedingung fiir u wurden inzwischen Eindeutigkeitsbeweise fiir Liisungen der Differentialgleichung (1) unter den Randbedingungen (2)(n und unter allgemeineren Randbedingungen(*) erbracht. C. SEYFERTH Physikalisch- Technischee Irwtitut Berlin
These results were confirmed by Nishiyamac3), who found moreover in similarly thin specimens the orientation relationship originally proposed by Bain, i.e.
t J
die Differentialgleichung (1) und die Randbedingungen (2). Diese Fur&ion hat die pathologische Eigenschaft
der Deutschen Akademie
Recently Pitschc2), working with test-pieces of such thinness that electron-diffraction patterns could be obtained, pointed out that a different orientation relationship was present, i.e.
der Wissenschuften
Nishiyama remarks, however, that the accuracy of his electron-diffraction patterns is not more than 5”. This is, indeed, insufficient to state with certainty whether the Pitsch or the Bsin relation is actually present, for this inaccuracy is of the same order of magnitude as the difference in orientation between the three relations mentioned above. 2. In view of the interesting suggestion made by Bain(*) that the transition of f.c.c. austenite into b.c.c. martensite might be brought about by a simple compression parallel to one of the cube axes of the originally face-centred cubic lattice, we prepared iron-nickel single crystals with a composition of 30.2 wt.% Ni and subjected them to a compressive force at room temperature. The compression experiments were carried out with circular discs with a t This work is part of the program of the Dutch Foundation of Fundamental Research (F.O.M.-T.N.O.), research group Prof. Dr. W. G. Burgers.