- Email: [email protected]
Segregation of solute atoms during strain aging
SEGREGATION T. MURAt,
OF SOLUTE
ATOMS
E. A. LAUTENSCHLAGERt
DURING
STRAIN
and J. 0.
AGING+
BRITTAINt
A theory and verifying experimental dete are prezented to explain strain aging of iron end steel under zeverel sging strfbins,atressesand temperatures. The theory developes a relation between the number of solute atoms migrating to a dislocation end the aging time which eventumllyleads to saturation. A term to account for the bulk diffusion which is due to the carbon concentration gradient developed in the vicinity of the dislocation is also included. The experimental resulte show the relation between the increeae of the yield point cfter aging and the eging time. Both the theory and the experimental results d8er from Harper’s, Bullough and Newmzn ‘s end Hem’s formulas. SEGREGATION
D’ATOMES
SOLUTES
LORS DU VIEILLISSEMENT
DE DEFORMATION
Les autaurs pr6sentent une th4orie et des r&ultats concordants pour expliquer le vieilliswment de deformation du fer et de l’scier pour quelques deform&ions, tensions et temperatures. Lz th4orie developpe une relation entre le nombre des at&es en solution qui migrent vers une dislocation et le temps de vieillissement qui conduit Bventuellement A la saturation. Un facteur tenant compta de la diffusion glob&, due au gradient de la concentration du carbone, qui se fait au voisinage de la dislocation, est aussi in&is. Les r4sultats exp6rimenteux montrent la relation entre l’accroissement de la limite Blaetique (yield point) apms vieillissement et le temps de vieillissement. LB theOrie ainsi que les r&mltets exp&imentaux different des formules de Harper, Bullough, Newman et Hem. AUSSCHEIDUNG
GELOSTER
ATOME
WAHREND
DER RECKALTERUNG
Eine Theorie,zur E&l&rung der Reckalterung von Eizen und Stahl unter dem Einflull verschiedener Dehmmgen, Spanmmgen und Temper&u-en wird mitgeteilt und uber experimentelle Reeultate, die diese Theorie bestitigen, wird berichtet. Die Theorie entwickelt eine Beziehuug zwischen der Zahl der g&&en Atome, die zu einer Versetzuug wandern und der Alterungszeit, die zur Slittigung flihrt. Ein Glied, das einer Volumdiifusion Rechnung tr>, wird mitgefiihrt; dieses riihrt her von dem Konzentrationsgradientan des Kohlenstoffs in der N&he der Versetzung. Die experimentellen Ergebnisse zeigen den Zusammenhang zwischen der Zunahme der Streckgrenze nach dem Altern und der Al-it. Sowohl die Theorie, wie such die experimentellen Ergebnisse untarscheiden sich von den Formeln von H-r, Bullough und Newman und Hem.
1. INTRODUCTION In
0
previous paper (l) the authors
discussed
the
tiuenca of aging stress, aging time and testing temperature on the return of the yield point after 8 str8in aging tre8tment. Here they discuss more quantitcltively the degree of strrtin aging in terms of segregation of solute atoms to 8 dislocation during strain aging. The effect of aging stress 8nd prior strain on strain aging where the degree of strain aging is taken-as the magnitude of the return of the yield point is shown to be in good 8greement with the yield point criterie developed in the previous paper. The segregation of soIute etoms to an edge dislomtion is accounted for by the migr8tion due to the drift force after Cottrell end Bilbyc2) and the diffusion current due to concentration gradients. Both terms are included in the continuity equ8tion for the l The rezearchwas supported by the U.S. Air Force through the Air Force Of&e of Scienti3c Research of the Air Research end Development Command. Received May 9, 1900; revised September 13. 1960. 7 Department of Metallurgy Materials Science, Northwestern University, Evanston, Illinois. $ Remarch Department, Allis-Chalmers, Milwaukee, Wieconsin.
ACTA 5
METALLURGICA,
VOL.
9, MAY
1961
proposed model. Since the solution of the oontinuity equation presents some difficulties, we have employed the technique of constructing particular solutions for two extreme csses in order to obtain a reasonable approximation. The resulting solution has been compared with 8 rather extensive series of experimental dats and found to give reasonable agreement. Finally, we heve compared 8 number of the current theories of strain 8ging and have shown that our solution w8s to be preferred. 2. THE
EXPERIMENTAL
PROCEDURE
The m8terial used in the experiments ~8s ingot’iron (0.013 C, 0.0125 Mn, 0.623 Si, 0.0023 S, 0.11 Cu, 0.004 N). Tensile specimens which had 8 reduced croes section of 1.25 in. and 8 diameter of 0.1 in. were annealed at 700°C for 4 hr in 8 dry hydrogen atmosphere and furnace cooled. The heat treetment resulted in 8 gr8in size of 0.042 mm. Specimens were deformed in tension at the aging temperatures at 8 head speed of 0.01 in./mih. Following the pl8stio deformcttion the load w8s reduced to 8 predetermined stress level and the specimens were aged while under locld. The stress maintained on the specimens dming 453
ACTA
454
METALLURGICA,
VOL.
9, 1961
few 30’ c
l
30
'
10)
IO4
0.04
3000pr
.04 15000 .04 25000 .04 35000
30 30
l
tot
Strain Strew
4
IO'
10's.c
Fm. 1. The experimental date for the return of yield point and the aging time under several aging streaeea 6xed aging temperature and aging strain. The return of yield point is defined aa Aala,, where Aa = a” - q, U, ie the stress at the upper yield point and a, is the flow stress at a de&nated pre-plastic strain.
lot/T)% 0
50
ID0
I50
200
250
I#(cmi/*K)” 300
350
400
450
FIG. 2. The experimental data for the return of yield point and (ot/!Z’)*/* under eeveral aging temperatures, plastic straina and aging &eases, where D is the diffusion constant, t the aging time and T is the aging temperature. The dotted linee show saturated va.luea of Au/u, predicted by Harper’s formula.
eging wee designated as aging stress. The aging t.empe&ure veried from 0 to SOY!, the aging stress v8ried 3000-26,000 lb/in”, end the return of yield point w8s measured at plastic strains of 4-12 per cent. The degree of aging is denoted by Aala, where Au = a, - a,, a, is the stress at the upper yield point and a, is the flow stress at a designated preplastic str8in.
3. EXPERIMENTAL
RESULTS
The strain aging curves shown in Fig. 1 are typical of the results obtained at 0,20,30,40 and 60% which show the influence of aging time and stress upon the degree of strain aging at a constant aging temperature and strain. The experimental points plotted in Figs. 1 and 2 were obtained from 8 least squere line drawn through the measured Aala, vs. plastic str8in at 8
MURA
et al.:
SEGREUATION
TABLE 1
= Aging temp. (“C)
1d%2cing % n
2 :
Strain
-0.08 0.08 0.12 0.04 0.04 0.04 0.04 0.04
Aging stress
(lb/in') 25,000 25,000 25,poo 15,000 25,000 25,000 3,000 15,000
constant aging time, temperature end aging stress similar to the procedure described previously.(3) For aging times of less than 3 hr data from 3 to 5 specimens are included in the least square calculation while for aging times of 8 hr or more the plotted points are the average of the measured values of A+, on two or more specimens et the indicated plastic strain end aging time, temperature and stress. In Fig. 2 the abscissa is (Dt/!Q2j3 x 1012 where D = 0.02 exp (-20,000/RT)(4) is the diffusion coefficient, T is the aging temperature and t is the aging time. The various points are identified with the aging variebles listed in Table 1. Note that the dotted lines in Fig. 2 are the saturated value for strain aging as predicted by Harper’s formula, the derivation of this saturation limit will be discussed in a later section. A comparison of the curves A and A or x and v in Fig. 2 shows that the degree of strain aging Au/a, increased as the aging stress increased. This effect of the aging stress is perhaps somewhat better shown in F‘ig. 1. The comparison of the curves for x, l and IJ in Fig. 2 shows that the ‘degree of strain aging at & constant aging temperature and stress decreases with an increase in the plastic strain. This observation has been previously noted.@) Although the abscissa was taken as (B/T) 2J3,the cancellation of the effect of aging the temperature was not complete (compare o and 0). 4. DISCUSSION
In a previous paper, Mura and Brittrtin(f) proposed that the magnitude of the yield point after aging increased with an increase of the parameter, PAN/ Bps, where L is the average distance of the movable part of the dislocation, A the interaction constint between the dislocation and a solute atom, N the number of solute atoms in the atmosphere per unit length of the dislocation, S the line tension and p the radius of the atmosphere. They discussed the dependency of S on the aging stress and concluded that an increase of the aging stress resulted in a decrease in S and consequently the parameter L2AN/Sp8 increased, and this agrees with
DURING
STRAIN
AGINU
45s
the first mentioned observation noted in Figs. 1 and 2. The increase of the aging strain results in a decrease of L, because the movable part of the disloccltion is decreased by trees or piled-up dislocations. The decreasing of L decreases the parameter ,L2AN/Sps and results in decreasing of the degree of the aging. This is in agreement with the second mentioned observation of the present experiments. The only aging time variable is N. According to Cottrell and Bilbyf2) N increases with the aging time according to a t2J3law. But the t2i3 law does not allow for saturation due to the balance between the concentration gradient and the drift flow towards its dislocation. Harper’@ has proposed a generalization of their results which attempts to account for the competition between adjacent dislocations. However, Harper’s proposal does not compensate for the neglect of the diffusion current due to concentration gradients. Papers by Hamt6), and Bullough snd Newman(v) have considered the effect of the diffusion current. The two papers used different boundary conditions snd obtained different results. The concentration gradient of solute atoms retards the &ing process in Bullough and Newman’s theory; on the other hand, it assists the process in Ham’s theory. The authors propose a simpler theory in the next section and compare it to the other theories in the last section with the experimental results presented in Figs. 1 and 2. It should be noted that the additional experimental data obtained at 0 and 20’ show exactly the same trends described in Figs. 1 and 2 but have not been included in the present paper since the tests were aborted due to the extremely long times required to approach saturation at these temperatures. The solution of the $i&%.sion equatiolz The model employed in describing the movement of solute atoms is that en edge dislocation lies along the z-axis and that the migration of solute atoms occurs in the x-y plane. If I is the concentration of solute atoms, the equation of continuity of solute atoms citn be written as
while V is the interaction energy between an edge dislocation and a solute atom: J’ = 4//(x2 and k is Boltzman’s constant, temperature.
+ y2)
(2)
T the absolute aging
ACTA
456 The
above
METALLURQICA,
equation oan be written in a simpler
VOL.
9, 1961 I
I
form :
aA @)
(3)
by the transformation 4 = -(A/WYW
+ Y%
?iJ= -(A/Wq(zS
+ y”),
(4
7 = (Dk*!P/Ae)t. It is assumed that the dislocation is isolated, that the concentration of solute atoms can not exceed a certain value 2i (saturated value of 1) and the initial value is $. The initial condition is at t = 0.
A=&
(5)
The boundary conditions are 1 = &at infinity, i.e.
q5= 0
and
w= 0
(6)
J = 2i in the effective atmosphere.
(7) The effective atmosphere is defined as the domain where the concentration of solute atoms has the saturated value I,. The domain corresponds to the precipitated domain in Ham’s theory or the core in Bullough and Newman’s theory. The effective atmosphere is created below the edge dislocation and Gnally becomes a circle 4 = A. The center of the oircle 4 = rpr is at x = 0, and y = -y,/2 where K = (A/WI& (8) It can be seen that a combined solution of the stationary solution and the drift solution for, the two extreme cases is:
xu
(9)
[ T-
s 6
where a[ ] = 1
when
T>
n[ ] = 0
when
7 <
_,(P
s
+ f)-”
+ _m($* + I)-”
d+ d# (10)
which satisfies the diffusion equation (3) everywhere except on a curve C and satisfies the initial condition (6) and the boundary conditions (6) and (7). The curve C is determined by 4 and v satisfying the relation : 7= s
d _mw
+
lu”r8 a+
(11)
effective atmosphere in the boundary condition (7) is the domain between the circle 4 = & and the curve C. The sketch of the solution (9) for a given t is shown in Fig. 3. The curve C is initially a point at
l----L
Fm. 3. Sketch of the distribution of the concentration of solute atoms 1 at an arbitrary time t. 1 has values of & outside the curve-C, A,[1 + (A,/&, - l)(exp + - l)/ (exp I#, - I)] within curve C and A1 in the region between curve C and the circle 4 = +1 (the effective etmosphere). The dashed curve representi the curve c after a time interval of At.
the origin of co-ordinates and expands with an increase in the time as shown by the dotted line in Fig. 3. The effective atmosphere becomes the circle 4 = h after inilnite time. It should be noted that the solution (9) gives the Maxwell distribution outside the atmosphere after an infinite time. The degree of aging after an aging time is proportionate to the number of solute atoms N that have migrated to the dislocations. Since the region of highest solute content is the effective atmosphere, we can take the magnitude of the strain aging to be proportionate to the area of the effective atmosphere. Let the domain area be F, where F is a function of aging time t. The maximum value of F is the area of the circle 4 = A at the stationary state, and this corresponds to the saturation value of the degree of aging. Let A which is equal to N/(N)_ represent the degree of segregation of solute atoms in terms of the saturated valued of solute atoms at the dislocations. Then we have
A = NI&,m, = WW,,, or in terms of solutions (9) and (11) (V + w”)-” a+ +J
The
where y1 is defined Corn
= F/(FL,
MURA
et al.:
SEGREGATION
Fro. 4. The curvea represent the resent theory for the &sin aging, where A = Au/(I; u)m.x. The points repent the experiments1 observations of the return of yield point in Fig. 2, where K = (2A/7rkyy,‘)‘~’~88 t&en 1010(0K/cm*)*/8.
DURING
STRAIN
457
AGING
A K* IOm(*K/cm’)a’a
0.2
. (Dt/l)-K
0
IO
20
30
40
SO
60
FIG. 5. The magnitudes of the alopes of experimental curves in Fig. 2 are plotted against Au/u,. The lines formed by extending the initial straight portion of the curves correspond to the linearity predicted by Harper’s formula.
A
.04
0
.06
.I2
.I6
.20
.24
26
Equation (12) can be expreesed in terms of r by equation (13) A w (4+)e’3 A M 1-
;
- ;
(2+r)-“3
(45/w) + - - * for? +
;
(2+/n)-’
(14)
+
where 7 = Th” = (ot/T)(A/&3).
(15)
Equation (14) is shown in Fig. 4 where K = (2A/7rky,3)a’3 and the data in Fig. 2 have been plotted for R = 10ro(K”/cma)a~3, i.e. for A = 3 x 10e80 erg cm and y1 = 30 A. The value of 30 A for the diameter of the saturated effective atmosphere, yi, is not an unreasonable value. Since A = Aa/(A = N/(N),,, ie independent of the aging strain and stress, and only depends on the aging time, all data under several aging conditions should fit one curve. Although the fit of the experimental data to the theoretical curve, Fig. 4, shows some scatter at large A, the present theory had a better fit than the previously mentioned theories.f6’)
co?wparicum oftheories The comparison of the four theories and the experimental reaulte requires a determination of (Au),, and the other unknown physical conetanta which
appear in each theory. While the experimental deterie in principle a eimple matter, mination of (Aa),, unfortunately under the conditions of our experimenta it requirea such long set up times on the Instron that we were not able to pursue this matter. Since the estimations of these quantitiea involve some uncertaint&, it is desirable to compare the theories without using these quantities. Also, eince there may be some question as to whether our data, which was obtained under carefully controlled conditiona and with good experimental technique, or in fact any experimental data is sensitive enough to be used to substantiate one of the theoriee by a direct comparison of theory and observation, we have adopted the procedure of comparing the theories in their differential form with the differential form of the experimental data aa depicted iri Fig. 5. This technique which does not require the UBBof the unknown physical con&ante of the theories enabled us to quickly discard two of the theories and to find a better agreement between one of the theoriea and the experimental results. The slopes, lo-l2 d(Au/u,)/d(Dt/T)“3, of the strain aging curves were obtained from Fig. 2 and plotted in Fig. 5 against the abaciesa Aala,. It can be seen that the relation between dA/dP and A is initially linear but deviates from a straight line aa strain aging approaches a titurated value. Since
ACTA
458
METALLURGICA,
VOL.
9, 1961
0.7
Q6
A: (I/K)dA/d(Dt/T)*”
h
9:
al
Harper’s formula, differential
A= 1-
form is & linear
0.2
0.3
dA/d(Dt/ r’)*” .
Prerent
Theory
6ullwgh and Newman
05 0.6 0.7 0.0 A FIQ. 6. The slope r&&ions between dA/dW end A are shown with curves A for present theory, curve B for Bullough and Ne-n’s theory and 0 for Ham’s theory.
exp
0.4
f--k(tD/~)~/~], in the dA/d(tD/T)2/3 =
relation
k(1 - A), the linear part of the curves in Fig. 5 agrees with Harper’s theory but it deviates from his theory as the saturation is approached. The extension (dotted lines) of the linear parts of the curves in Fig. 5 gives the saturated values of ha/a, predicted by Harper’s theory. The saturated values are superimposed on the experimental curves in Fig. 2 as dotted lines. It is clear that the experimental curves have not saturated at the values predicted by Harper’s theop but continue to increase with increasing time. The results of the present theory, Bullough and Newman’s, and Ham’s theories are compared with the experimental observation in Fig. 5 by plotting the relations between dA~dPfs and A in Fig. 6 with curves A for the present theory, curve B for Bullough and Newman’s theory and C for Ham’s theory. It is clear that Ham’s theory does not agree with the characteristic of Fig. 5. Curve B deviated from the straight line too early compared with the experimental observation. The present theory has a more reasonable agreement with the experimental characteristics. Although Ham noted the discrepanoy between his theory and the experimental results, Bullough and Newman’s treatment gave good agreement with Rockwell hardness data(*) on the strain aging of low carbon steel. Bullough and Newman treated only the edge dislocation, but the atmosphere of solute atoms in their theory is around the dislocation (a circle with
the dislocation at the center). This appears to be permissible however only for screw dislocations,@) for when the dislocation is of the edge type the atmosphere should be in a smeared out region either above or below the dislocation in order to minimize the strain energy and thus will be dependent upon whether the_solute atom causes an expansion or contraction in the lattice of the solvent. A second difliculty in their theory is that the concentration of solute atoms reached after a long aging period (the steady state distribution of solute atoms around the dislocations) is larger everywhere than the initial concentration of solute atoms contrary to the conservation law of matter. The present theory has no such contradictions and gives as good or better agreements with the experimental observations, however the mathematical techniques used to obtain the tlnal solution sacrificed some of the rigor of the mathematics. REFERENCES
i: 3. 4. :: 7. 8. 9.
T. MURA end J. 0. BIU~AIN, AC& Met. 8, 767 (1969). A. H. CO~ELL and B. A. BILBY, Proc. Phys. Sm., Lad. Ai%& 29 (1949). J. 0. BRITTAIN and S. E. BRONISZ, Tram. Amer. Inet. Min. (Met&!.) Engm. 218, 289 (1960). R. H. DOREMUB,AC&ZMet. 7, 398 (1969). S. HAXPEIZ,Phye. Reu. 88, 709 (1961). F. 8. Hnar, J. A@. Phya. 80, 916 (1969). Proc. Roy. Soo., Land. R. Burmmm end R. C. NE-, A%49, 427 (1969). E. S. DAVENPOBT and E. C. BUN, Tram. Aw. Sot. Met& a8, 1047 (1935). A. W. COCXIAXDT,G. Scaosax end H. WIDEB~IOE. Ada Met. 8, 633 (1966).
OF SOLUTE
ATOMS
E. A. LAUTENSCHLAGERt
DURING
STRAIN
and J. 0.
AGING+
BRITTAINt
A theory and verifying experimental dete are prezented to explain strain aging of iron end steel under zeverel sging strfbins,atressesand temperatures. The theory developes a relation between the number of solute atoms migrating to a dislocation end the aging time which eventumllyleads to saturation. A term to account for the bulk diffusion which is due to the carbon concentration gradient developed in the vicinity of the dislocation is also included. The experimental resulte show the relation between the increeae of the yield point cfter aging and the eging time. Both the theory and the experimental results d8er from Harper’s, Bullough and Newmzn ‘s end Hem’s formulas. SEGREGATION
D’ATOMES
SOLUTES
LORS DU VIEILLISSEMENT
DE DEFORMATION
Les autaurs pr6sentent une th4orie et des r&ultats concordants pour expliquer le vieilliswment de deformation du fer et de l’scier pour quelques deform&ions, tensions et temperatures. Lz th4orie developpe une relation entre le nombre des at&es en solution qui migrent vers une dislocation et le temps de vieillissement qui conduit Bventuellement A la saturation. Un facteur tenant compta de la diffusion glob&, due au gradient de la concentration du carbone, qui se fait au voisinage de la dislocation, est aussi in&is. Les r4sultats exp6rimenteux montrent la relation entre l’accroissement de la limite Blaetique (yield point) apms vieillissement et le temps de vieillissement. LB theOrie ainsi que les r&mltets exp&imentaux different des formules de Harper, Bullough, Newman et Hem. AUSSCHEIDUNG
GELOSTER
ATOME
WAHREND
DER RECKALTERUNG
Eine Theorie,zur E&l&rung der Reckalterung von Eizen und Stahl unter dem Einflull verschiedener Dehmmgen, Spanmmgen und Temper&u-en wird mitgeteilt und uber experimentelle Reeultate, die diese Theorie bestitigen, wird berichtet. Die Theorie entwickelt eine Beziehuug zwischen der Zahl der g&&en Atome, die zu einer Versetzuug wandern und der Alterungszeit, die zur Slittigung flihrt. Ein Glied, das einer Volumdiifusion Rechnung tr>, wird mitgefiihrt; dieses riihrt her von dem Konzentrationsgradientan des Kohlenstoffs in der N&he der Versetzung. Die experimentellen Ergebnisse zeigen den Zusammenhang zwischen der Zunahme der Streckgrenze nach dem Altern und der Al-it. Sowohl die Theorie, wie such die experimentellen Ergebnisse untarscheiden sich von den Formeln von H-r, Bullough und Newman und Hem.
1. INTRODUCTION In
0
previous paper (l) the authors
discussed
the
tiuenca of aging stress, aging time and testing temperature on the return of the yield point after 8 str8in aging tre8tment. Here they discuss more quantitcltively the degree of strrtin aging in terms of segregation of solute atoms to 8 dislocation during strain aging. The effect of aging stress 8nd prior strain on strain aging where the degree of strain aging is taken-as the magnitude of the return of the yield point is shown to be in good 8greement with the yield point criterie developed in the previous paper. The segregation of soIute etoms to an edge dislomtion is accounted for by the migr8tion due to the drift force after Cottrell end Bilbyc2) and the diffusion current due to concentration gradients. Both terms are included in the continuity equ8tion for the l The rezearchwas supported by the U.S. Air Force through the Air Force Of&e of Scienti3c Research of the Air Research end Development Command. Received May 9, 1900; revised September 13. 1960. 7 Department of Metallurgy Materials Science, Northwestern University, Evanston, Illinois. $ Remarch Department, Allis-Chalmers, Milwaukee, Wieconsin.
ACTA 5
METALLURGICA,
VOL.
9, MAY
1961
proposed model. Since the solution of the oontinuity equation presents some difficulties, we have employed the technique of constructing particular solutions for two extreme csses in order to obtain a reasonable approximation. The resulting solution has been compared with 8 rather extensive series of experimental dats and found to give reasonable agreement. Finally, we heve compared 8 number of the current theories of strain 8ging and have shown that our solution w8s to be preferred. 2. THE
EXPERIMENTAL
PROCEDURE
The m8terial used in the experiments ~8s ingot’iron (0.013 C, 0.0125 Mn, 0.623 Si, 0.0023 S, 0.11 Cu, 0.004 N). Tensile specimens which had 8 reduced croes section of 1.25 in. and 8 diameter of 0.1 in. were annealed at 700°C for 4 hr in 8 dry hydrogen atmosphere and furnace cooled. The heat treetment resulted in 8 gr8in size of 0.042 mm. Specimens were deformed in tension at the aging temperatures at 8 head speed of 0.01 in./mih. Following the pl8stio deformcttion the load w8s reduced to 8 predetermined stress level and the specimens were aged while under locld. The stress maintained on the specimens dming 453
ACTA
454
METALLURGICA,
VOL.
9, 1961
few 30’ c
l
30
'
10)
IO4
0.04
3000pr
.04 15000 .04 25000 .04 35000
30 30
l
tot
Strain Strew
4
IO'
10's.c
Fm. 1. The experimental date for the return of yield point and the aging time under several aging streaeea 6xed aging temperature and aging strain. The return of yield point is defined aa Aala,, where Aa = a” - q, U, ie the stress at the upper yield point and a, is the flow stress at a de&nated pre-plastic strain.
lot/T)% 0
50
ID0
I50
200
250
I#(cmi/*K)” 300
350
400
450
FIG. 2. The experimental data for the return of yield point and (ot/!Z’)*/* under eeveral aging temperatures, plastic straina and aging &eases, where D is the diffusion constant, t the aging time and T is the aging temperature. The dotted linee show saturated va.luea of Au/u, predicted by Harper’s formula.
eging wee designated as aging stress. The aging t.empe&ure veried from 0 to SOY!, the aging stress v8ried 3000-26,000 lb/in”, end the return of yield point w8s measured at plastic strains of 4-12 per cent. The degree of aging is denoted by Aala, where Au = a, - a,, a, is the stress at the upper yield point and a, is the flow stress at a designated preplastic str8in.
3. EXPERIMENTAL
RESULTS
The strain aging curves shown in Fig. 1 are typical of the results obtained at 0,20,30,40 and 60% which show the influence of aging time and stress upon the degree of strain aging at a constant aging temperature and strain. The experimental points plotted in Figs. 1 and 2 were obtained from 8 least squere line drawn through the measured Aala, vs. plastic str8in at 8
MURA
et al.:
SEGREUATION
TABLE 1
= Aging temp. (“C)
1d%2cing % n
2 :
Strain
-0.08 0.08 0.12 0.04 0.04 0.04 0.04 0.04
Aging stress
(lb/in') 25,000 25,000 25,poo 15,000 25,000 25,000 3,000 15,000
constant aging time, temperature end aging stress similar to the procedure described previously.(3) For aging times of less than 3 hr data from 3 to 5 specimens are included in the least square calculation while for aging times of 8 hr or more the plotted points are the average of the measured values of A+, on two or more specimens et the indicated plastic strain end aging time, temperature and stress. In Fig. 2 the abscissa is (Dt/!Q2j3 x 1012 where D = 0.02 exp (-20,000/RT)(4) is the diffusion coefficient, T is the aging temperature and t is the aging time. The various points are identified with the aging variebles listed in Table 1. Note that the dotted lines in Fig. 2 are the saturated value for strain aging as predicted by Harper’s formula, the derivation of this saturation limit will be discussed in a later section. A comparison of the curves A and A or x and v in Fig. 2 shows that the degree of strain aging Au/a, increased as the aging stress increased. This effect of the aging stress is perhaps somewhat better shown in F‘ig. 1. The comparison of the curves for x, l and IJ in Fig. 2 shows that the ‘degree of strain aging at & constant aging temperature and stress decreases with an increase in the plastic strain. This observation has been previously noted.@) Although the abscissa was taken as (B/T) 2J3,the cancellation of the effect of aging the temperature was not complete (compare o and 0). 4. DISCUSSION
In a previous paper, Mura and Brittrtin(f) proposed that the magnitude of the yield point after aging increased with an increase of the parameter, PAN/ Bps, where L is the average distance of the movable part of the dislocation, A the interaction constint between the dislocation and a solute atom, N the number of solute atoms in the atmosphere per unit length of the dislocation, S the line tension and p the radius of the atmosphere. They discussed the dependency of S on the aging stress and concluded that an increase of the aging stress resulted in a decrease in S and consequently the parameter L2AN/Sp8 increased, and this agrees with
DURING
STRAIN
AGINU
45s
the first mentioned observation noted in Figs. 1 and 2. The increase of the aging strain results in a decrease of L, because the movable part of the disloccltion is decreased by trees or piled-up dislocations. The decreasing of L decreases the parameter ,L2AN/Sps and results in decreasing of the degree of the aging. This is in agreement with the second mentioned observation of the present experiments. The only aging time variable is N. According to Cottrell and Bilbyf2) N increases with the aging time according to a t2J3law. But the t2i3 law does not allow for saturation due to the balance between the concentration gradient and the drift flow towards its dislocation. Harper’@ has proposed a generalization of their results which attempts to account for the competition between adjacent dislocations. However, Harper’s proposal does not compensate for the neglect of the diffusion current due to concentration gradients. Papers by Hamt6), and Bullough snd Newman(v) have considered the effect of the diffusion current. The two papers used different boundary conditions snd obtained different results. The concentration gradient of solute atoms retards the &ing process in Bullough and Newman’s theory; on the other hand, it assists the process in Ham’s theory. The authors propose a simpler theory in the next section and compare it to the other theories in the last section with the experimental results presented in Figs. 1 and 2. It should be noted that the additional experimental data obtained at 0 and 20’ show exactly the same trends described in Figs. 1 and 2 but have not been included in the present paper since the tests were aborted due to the extremely long times required to approach saturation at these temperatures. The solution of the $i&%.sion equatiolz The model employed in describing the movement of solute atoms is that en edge dislocation lies along the z-axis and that the migration of solute atoms occurs in the x-y plane. If I is the concentration of solute atoms, the equation of continuity of solute atoms citn be written as
while V is the interaction energy between an edge dislocation and a solute atom: J’ = 4//(x2 and k is Boltzman’s constant, temperature.
+ y2)
(2)
T the absolute aging
ACTA
456 The
above
METALLURQICA,
equation oan be written in a simpler
VOL.
9, 1961 I
I
form :
aA @)
(3)
by the transformation 4 = -(A/WYW
+ Y%
?iJ= -(A/Wq(zS
+ y”),
(4
7 = (Dk*!P/Ae)t. It is assumed that the dislocation is isolated, that the concentration of solute atoms can not exceed a certain value 2i (saturated value of 1) and the initial value is $. The initial condition is at t = 0.
A=&
(5)
The boundary conditions are 1 = &at infinity, i.e.
q5= 0
and
w= 0
(6)
J = 2i in the effective atmosphere.
(7) The effective atmosphere is defined as the domain where the concentration of solute atoms has the saturated value I,. The domain corresponds to the precipitated domain in Ham’s theory or the core in Bullough and Newman’s theory. The effective atmosphere is created below the edge dislocation and Gnally becomes a circle 4 = A. The center of the oircle 4 = rpr is at x = 0, and y = -y,/2 where K = (A/WI& (8) It can be seen that a combined solution of the stationary solution and the drift solution for, the two extreme cases is:
xu
(9)
[ T-
s 6
where a[ ] = 1
when
T>
n[ ] = 0
when
7 <
_,(P
s
+ f)-”
+ _m($* + I)-”
d+ d# (10)
which satisfies the diffusion equation (3) everywhere except on a curve C and satisfies the initial condition (6) and the boundary conditions (6) and (7). The curve C is determined by 4 and v satisfying the relation : 7= s
d _mw
+
lu”r8 a+
(11)
effective atmosphere in the boundary condition (7) is the domain between the circle 4 = & and the curve C. The sketch of the solution (9) for a given t is shown in Fig. 3. The curve C is initially a point at
l----L
Fm. 3. Sketch of the distribution of the concentration of solute atoms 1 at an arbitrary time t. 1 has values of & outside the curve-C, A,[1 + (A,/&, - l)(exp + - l)/ (exp I#, - I)] within curve C and A1 in the region between curve C and the circle 4 = +1 (the effective etmosphere). The dashed curve representi the curve c after a time interval of At.
the origin of co-ordinates and expands with an increase in the time as shown by the dotted line in Fig. 3. The effective atmosphere becomes the circle 4 = h after inilnite time. It should be noted that the solution (9) gives the Maxwell distribution outside the atmosphere after an infinite time. The degree of aging after an aging time is proportionate to the number of solute atoms N that have migrated to the dislocations. Since the region of highest solute content is the effective atmosphere, we can take the magnitude of the strain aging to be proportionate to the area of the effective atmosphere. Let the domain area be F, where F is a function of aging time t. The maximum value of F is the area of the circle 4 = A at the stationary state, and this corresponds to the saturation value of the degree of aging. Let A which is equal to N/(N)_ represent the degree of segregation of solute atoms in terms of the saturated valued of solute atoms at the dislocations. Then we have
A = NI&,m, = WW,,, or in terms of solutions (9) and (11) (V + w”)-” a+ +J
The
where y1 is defined Corn
= F/(FL,
MURA
et al.:
SEGREGATION
Fro. 4. The curvea represent the resent theory for the &sin aging, where A = Au/(I; u)m.x. The points repent the experiments1 observations of the return of yield point in Fig. 2, where K = (2A/7rkyy,‘)‘~’~88 t&en 1010(0K/cm*)*/8.
DURING
STRAIN
457
AGING
A K* IOm(*K/cm’)a’a
0.2
. (Dt/l)-K
0
IO
20
30
40
SO
60
FIG. 5. The magnitudes of the alopes of experimental curves in Fig. 2 are plotted against Au/u,. The lines formed by extending the initial straight portion of the curves correspond to the linearity predicted by Harper’s formula.
A
.04
0
.06
.I2
.I6
.20
.24
26
Equation (12) can be expreesed in terms of r by equation (13) A w (4+)e’3 A M 1-
;
- ;
(2+r)-“3
(45/w) + - - * for? +
;
(2+/n)-’
(14)
+
where 7 = Th” = (ot/T)(A/&3).
(15)
Equation (14) is shown in Fig. 4 where K = (2A/7rky,3)a’3 and the data in Fig. 2 have been plotted for R = 10ro(K”/cma)a~3, i.e. for A = 3 x 10e80 erg cm and y1 = 30 A. The value of 30 A for the diameter of the saturated effective atmosphere, yi, is not an unreasonable value. Since A = Aa/(A = N/(N),,, ie independent of the aging strain and stress, and only depends on the aging time, all data under several aging conditions should fit one curve. Although the fit of the experimental data to the theoretical curve, Fig. 4, shows some scatter at large A, the present theory had a better fit than the previously mentioned theories.f6’)
co?wparicum oftheories The comparison of the four theories and the experimental reaulte requires a determination of (Au),, and the other unknown physical conetanta which
appear in each theory. While the experimental deterie in principle a eimple matter, mination of (Aa),, unfortunately under the conditions of our experimenta it requirea such long set up times on the Instron that we were not able to pursue this matter. Since the estimations of these quantitiea involve some uncertaint&, it is desirable to compare the theories without using these quantities. Also, eince there may be some question as to whether our data, which was obtained under carefully controlled conditiona and with good experimental technique, or in fact any experimental data is sensitive enough to be used to substantiate one of the theoriee by a direct comparison of theory and observation, we have adopted the procedure of comparing the theories in their differential form with the differential form of the experimental data aa depicted iri Fig. 5. This technique which does not require the UBBof the unknown physical con&ante of the theories enabled us to quickly discard two of the theories and to find a better agreement between one of the theoriea and the experimental results. The slopes, lo-l2 d(Au/u,)/d(Dt/T)“3, of the strain aging curves were obtained from Fig. 2 and plotted in Fig. 5 against the abaciesa Aala,. It can be seen that the relation between dA/dP and A is initially linear but deviates from a straight line aa strain aging approaches a titurated value. Since
ACTA
458
METALLURGICA,
VOL.
9, 1961
0.7
Q6
A: (I/K)dA/d(Dt/T)*”
h
9:
al
Harper’s formula, differential
A= 1-
form is & linear
0.2
0.3
dA/d(Dt/ r’)*” .
Prerent
Theory
6ullwgh and Newman
05 0.6 0.7 0.0 A FIQ. 6. The slope r&&ions between dA/dW end A are shown with curves A for present theory, curve B for Bullough and Ne-n’s theory and 0 for Ham’s theory.
exp
0.4
f--k(tD/~)~/~], in the dA/d(tD/T)2/3 =
relation
k(1 - A), the linear part of the curves in Fig. 5 agrees with Harper’s theory but it deviates from his theory as the saturation is approached. The extension (dotted lines) of the linear parts of the curves in Fig. 5 gives the saturated values of ha/a, predicted by Harper’s theory. The saturated values are superimposed on the experimental curves in Fig. 2 as dotted lines. It is clear that the experimental curves have not saturated at the values predicted by Harper’s theop but continue to increase with increasing time. The results of the present theory, Bullough and Newman’s, and Ham’s theories are compared with the experimental observation in Fig. 5 by plotting the relations between dA~dPfs and A in Fig. 6 with curves A for the present theory, curve B for Bullough and Newman’s theory and C for Ham’s theory. It is clear that Ham’s theory does not agree with the characteristic of Fig. 5. Curve B deviated from the straight line too early compared with the experimental observation. The present theory has a more reasonable agreement with the experimental characteristics. Although Ham noted the discrepanoy between his theory and the experimental results, Bullough and Newman’s treatment gave good agreement with Rockwell hardness data(*) on the strain aging of low carbon steel. Bullough and Newman treated only the edge dislocation, but the atmosphere of solute atoms in their theory is around the dislocation (a circle with
the dislocation at the center). This appears to be permissible however only for screw dislocations,@) for when the dislocation is of the edge type the atmosphere should be in a smeared out region either above or below the dislocation in order to minimize the strain energy and thus will be dependent upon whether the_solute atom causes an expansion or contraction in the lattice of the solvent. A second difliculty in their theory is that the concentration of solute atoms reached after a long aging period (the steady state distribution of solute atoms around the dislocations) is larger everywhere than the initial concentration of solute atoms contrary to the conservation law of matter. The present theory has no such contradictions and gives as good or better agreements with the experimental observations, however the mathematical techniques used to obtain the tlnal solution sacrificed some of the rigor of the mathematics. REFERENCES
i: 3. 4. :: 7. 8. 9.
T. MURA end J. 0. BIU~AIN, AC& Met. 8, 767 (1969). A. H. CO~ELL and B. A. BILBY, Proc. Phys. Sm., Lad. Ai%& 29 (1949). J. 0. BRITTAIN and S. E. BRONISZ, Tram. Amer. Inet. Min. (Met&!.) Engm. 218, 289 (1960). R. H. DOREMUB,AC&ZMet. 7, 398 (1969). S. HAXPEIZ,Phye. Reu. 88, 709 (1961). F. 8. Hnar, J. A@. Phya. 80, 916 (1969). Proc. Roy. Soo., Land. R. Burmmm end R. C. NE-, A%49, 427 (1969). E. S. DAVENPOBT and E. C. BUN, Tram. Aw. Sot. Met& a8, 1047 (1935). A. W. COCXIAXDT,G. Scaosax end H. WIDEB~IOE. Ada Met. 8, 633 (1966).