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Strain-hardening of spheroidized high carbon steels
STRAIN-HARDENING OF SPHEROIDIZED HIGH CARBON STEELS L. ;\S.%SD*
and J. GCRL.AND
Division of Engineering. Brown University. Providence. RI. U.S.A.
Abstract-The stmin-hardening behavior of spheroidized steels is examined in terms of continuum and quasi-continuum models based on the requirement that continuit! be ~in~ined at various boundaries in the two-phase pol>crystaIs. For plastic strains up to 3.5’,, the increase in the flow stress appears to by composed of contributions arising from (i) dislocations stored in the material for a) statistical and b) geometrical reasons and (ii) an average back stress predominantly due to the presence of the unrelaxed plastic strain discontinuity between the ferrite marrix and the cementite particles. The latter contribution. which is a large fraction of the total strain-hardening increment. increases rapidly with strain up to a plastic strain of 3.Y0 and remains approximately constant thereafter. The attainment of the maximum value of the back stress corresponds to the observed transition in the overall strain-hardening rate (“double-n” behavior). The c~perimentall\ determined components of the hardenine increment due to the geometrically necessary dislocations agree well with the predictions of Ashb& theorv. At strains greaier than 3.YJ0.the average back stress remains approuimatsly constant and the geometrically necessary dislocation density increases only slight@ with strsin. This indicates a deviation from Ashby’s theory which predicts a continuous increase of this dislocation population with strain. The strain-hardening increments at these strain levels therefore appear to be controlled main]! b> the increase in the statistical dislocation population.
R&m&On examine I’&rouissage d’acie:s i carbone sphCrdidal dsns des mod&s continus et quasicontinus qui reposent sur Is maintien de la continuiti aus divers in&aces des polycristaux biphases. Pour des dSformations plastiques allant jusqu’8 3.Y0. il semble que l’accroissement de la contrainte d’&oulement soit composi de contributions provenant (i) des dislocations smmagasinies dans It? materiau pour des raisons a) statistiques et b) giomttriques. et (ii) d’une contrainte interne due essentieilement B la prksence d’une discontinuiti de la d&formation plastiqus non relaxis entre la matrice ferritique et les particules de cementite. La derniere contribution. qui constitue une part importante de I’augmentation totale de durcissement, augmente rapidement avec la diformation jusqu.8 des defermations plastiques de 3.Y,,. puis reste pratiquement constante. Le moment oit la contrainte interrr atteint sa valeur maximale correspond 6 la transition observee dans la vitesse globale d‘ecrouissage. Les composantes exp&imentales de i’accroissement du durcissement dii aux dislocations geom&tiquement nkcessaires sont en bon accord avec Iss predictions dc la theorie
* ;Vow at Research Laboratory, U.S. Steel Corporation. 84onrotville. PA. U.S.A. 901
ANAND X?;D GURLAND:
902
STRAIN-HARDENING
OF SPHEROID~ED
Table 2. ~iicros~~ctural
KTRODUCI’ION X previous paper [l] considered the combined effects of cementite partides. grain boundaries and subgrain boundaries on the room temperature yielding behavior of spheroidized plain carbon steels. The spheroidized structures were obtained by austenitizing and quenching followed by either tempering at temperatures just under the A, temperature or thermal cycling about the ,-t, temperature. Electron microscopy revealed a subgrain-connects cementite particle distribution in the steeis which were spheroid~ed by quenching and tempering [l, 21, and a subgrain-free cementite particle distribution in large ferrite grains in the steels which were spheroidized by the quenchand-cycle procedure. The two different microstructural ty-pes were designated POB (particles on boundaries) and PIB (particles inside boundaries) respectively. In the POB specimens the subgrain size, E.t,,, stabilized by the particles. was found to control the lower yield stress via the relation G,. =
9.5
+
1.3%.$
’
(kgf/mm’)
11.3
+
parameters
POB specimens sp. No.
k.p (lrm)
Steel
f
C D” D
0.33 0.57 0.72 0.53 0.90 1.25
1.19 2.03 2.57 1.53 2.57 3.59
0.6 1 1.31
3 4 :
0.16 0.16 0.16 0.19 0.19
7
D
0.19
1.32
3.SO
2.01
(l)
(2)
(&
1
2
(&
k$ 2.14 t.59
PIB specimens
Sp. No.
Steel
I;
1 2 3 4
C C :
0.12 0.0s 0.15 0.19
0.26 0.36 0.54 0.92
0.52 1.69 1.2: 2.33
1.s-k 3.01 3.49 Il.41
:,
D
0.16 0.14
0.98 0.77
2.57 2.97
10.96 11.81
(1)
The looser yield stress of the PIB specimens was found to be predominantIy controlled by the ferrite grain size. i,. via the Hall-Petch relation Gy =
STEELS
1.87& ’ ” (kgf/mm’).
(2)
In PIB specimens the intraboundary particles at yielding contribute only a small strain-hardening term which increases the value of the friction stress (12.4kgfmm’) over that associated with grain boundary strengthening alone (8.8 f 0.8 kgf/mm’ [3]). It is the purpose of the present work to report the rest&s of an investigation of the strain-hardening behavior of spheroidized high carbon steels in both the quenched-and-ternary (POB) and quench~-andcycled (PIB) microstntctural conditions. An attempt is made to account for the various contributions to the stress increment of strain hardening on the basis of recent theories of particle and boundary strengthening. EXPERIMENTAL The chemical analyses of the two carbon steels are given in Table I. The heat treatment and metallographic procedures are described elsehwere [ 1,4]. Table 2 lists the volume fractioni the mean spherical particle dia.. 6 the mean free path between the particles of cementite. i,. the subgrain size, & (which
is the mean free path considering both particles and sub-sundries as barriers), for PO3 specimens, and the volume fraction,J, and particle dia.. Jr. of cementite particles lying within ferrite grains of mean linear intercept length i., for PIB specimens. Load-elongation curves in tension were obtained at room temperature with l-in. (2.54 x lo-’ m) gauge length, 0.25in. (4.4 x 10T3 m) dia. specimens on an Instron machine at a cross-head speed of O.O&in/min (8.4 x 10m6m’s). The load-elongation data were converted to true stress, G, true totat strain, E, and plastic strain E+,. where (31 EP = E - G/E. According to the Hollomon equation [5]. a=kE”,
(4)
the relation between ha and In E should be linear over the full range of strains, but, in our case, the data for each POB or PIB specimen may be approximated by two straight lines which intersect at about 5% strain. An example of this “double n” behavior is shown in Fig. I for PIB specimen No. 4 of Table 2. The calculated values of the variables and constants 42 4.1 r
b
Table 1. Composition of carbon steels Steel
C
c D
1.05 1.24
Composition* Mn Si 0.34 0.17
0.24 0.16
(wt.%) Al
P
S
0.008 0.009
0.012 0.012
0.011 0.013
* Chemical anal>-sis by Walter M. Saunders, Inc., Provi__
cience.
Kl.
I
-303
-343
-1 is
In*
Fig. 1. Double-n stmi~-hardening
behavior.
-223
n
of equation
(4 for both true total strains and plastic strains are reported in C-l].
-0
c
,.-
1
j-zE,_
”
s ‘M3C’_
‘?O
RESC-LTS .45-D DISCLSSIOS
r\rmstrong et nl. [6] have shown that at temperatures where grain boundary sliding and diffusion do not play a major role. the Aow stress. G,~. of a singlephase polycrystatline metal at a given plastic strain, fp. depends on the grain size. & through the rshtionship Gcp=
Gc,sp
+
ken i-9 ’ :,
f3,
where G~.~ I and k,, are experimental constants. In the case of spheroidized steels with subgrain boundaries pinned by cementite particles (POB). the yield strength is governed by the subgrain size (equation 1) [l]. and it may be assumed that the subgain boundaries behave in a manner analogous to grain boundaries also during strain-hardening. and
unfortunately, it is dif%cult to produce a large range of & for any one given steeI bq simple quenching and tempering. and an accurate determination of Gor, and kcP is not always possible. To obtain approximate estimates of these parameters, the data for steels C and D (which have volume fractions f equal to 0.16 and 0.19, respectively. i.e. values which are reasonably close to each other) were plotted together according to equation (6). and the dependence of the flow stress on the subgrain size is shown in Fig. 2. The values of G~,~, and kcP as determined by least squares anaiysis are listed in Table 3. T~~~~~~ stress ~10~~~.The flow stress at a strain E, is the result of strain induced hardening effects added to the initial yield strength at the very beginning of plastic deformation. The latter is given by equation (6) at zero plastic strain: G<,=()= 00,#,=0 -4”kQZO&’ 2.
(7)
The strain dependent hardening increment is accounted for by the dislocation density and by iongrange internal stresses opposing dislocation motion. The total dislocation density, at a particular value of the strain. is the sum of the statisticaily stored dislocation density. p’. and a geometrically necessary dislocation density. py [7]. The former results from chance encounters and mutual trapping of dislocations within the crystal. the latter is due to the requirement that continuity be maintained at internal boundaries of the polycrystal. The contribution Ao’ of the statistical dislocation density to the flow stress is assumed to be linearly superposed on A@, the con*Ths combined How stress increment should actually be the square root of the sum of the squares of Ati and brig. The above assumption of liner superposition makes The subsequent analysis more tractable.
x.2.P -i
,mm-+
Fig. 2. The relationship between the tiow stritss 0;” at two piastic strains, E,. and the subgram size. f.(.,
tribution of the geometrically necessary dislocation density [S].* Wilson and Monan [9] showed that. in a highcarbon steel spheroidized by quenching and tempering. unloading after tensile deformation left a residua1 stress system in the specimens such that compressive stresses in the ferrite were balanced by tensile stresses in the cementite. The interphase stress system increases rapidly during the first few percent of plastic extension and remains almost constant thereafter. The origin of these internal stresses has recent{> been considered theoretically on the basis of continuum models [lo. 1I]. and their presence is attributed to the unrelaxed plastic strain discontinuity M\veen the matrix and the elastically deforming parricles. The lesser contribution of the single phase pol!crystolline iron matrix to the internal back stresses may be neglected in steels with ;I large volume fraction of cementite. In spheroidized high carbon steels, therefore. the total strain in the matrix is subdivjded into two components. E; and EL, where l “pis the unrelasrd plastic strain discontinuity due to the particles and EL is the portion of the strain which has been relaxed by secondary dislocation processes [7. 11, 171. The unrelaxed strain gives rise to a hardening term. Aa’. which represents an average internal stress opposing Table 5. Calculated values of the constants c,~.<, and ke, in the Hall-Petch equation zSe = bO.cp- kS, ;.,i ’ for the POB specimens
904
ANAND
AND
GCRLAND:
S-IX~IN-HARDENING OF SPHEROIDIZED STEELS
dislocation motion. The relaxed strain portion determines the geometrically-necessary dislocation density and the resulting flow stress increment is. after Ashby [7]. Aa,, = (C, &-;
I,
(S)
where it is assumed (1) that the particle-pinned subgrain boundaries in the POB specimens are analogous to grain boundaries in their effect on generating a dislocation density [9], (2) that the particles uniformly affect the strain continuity at the subgrain boundaries (the constant C is then a function of the volume fraction of the second phase), and (3) that the geometrically necessary dislocations so generated are uniformly distributed within the volumes of the small subgrains under consideration. In summary, the flow stress of POB steels is given by: 0cp = 6cp=,, + do, + A& + Ao’
(9)
or in terms of equation (6):
50
;
40
E \
fy
I
SYMBOL
STEEL
c
.
0
.
+ 400
1
m x 30
-
.
ucs’o = l.O7i.,T,L2.
(11)
The zero value of the intercept is somewhat difficult to rationalize. Liu et al. [12] showed that the lattice friction stress in steels at the onset of plastic flow, as estimated by two alternative double-extrapolation techniques, is uO.~p=O= 3.66 kgf/mm’. Part of the discrepancy may be attributed to the limited data of Fig. 3. In view of the uncertainty, this quantity will be neglected with the realization that the following arguments are not affected significantly by the precise value of G~,~~=~. From equation (10) we may therefore write o,,~, = Aa’ + Agi.
(12)
There are (as yet) no simple geometrical arguments which can be used to calculate the fraction Ao’ of CJ~,~,arising from the statistical dislocation density p’, although it may be obtained from experimental measurements. The ideal statistical hardening curve is a Taylor-corrected stress-strain curve for a single crystal deforming by multiple slip or polyslip [13]. The strain-hardening behavior of a polycrystal with grain size much larger than the average slip distance of a single crystal, however. does approach the Taylor-corrected single-crystal strain-hardening behavior [S]. Therefore the data for a large-grained iron specimen (28Omm grain size. 0.02 wt.% impurity)
z
0
h bW
20
.
.
200
IO 20
I
26
1100
I
I
I
24
32
36
4:
44
-i ’ A:., ) mm-* Fig. 3. The How stress at zero plastic strain. G+=~. as a function of the subgrain size. il.;.
from the work of iMorrison [14] is used here to estimate Ao’ as a function of ep. by means of equation (4). The calculated values of A8 are plotted in Fig. 7. The average internal stresses Ao’ may now be calculated from AC+= r~,,~, - Ad
71~~r@cts of strain on CJ,,(~. The values of a,,= ,, were calculated from the extrapolation of the initial portion of the homogeneous part of the stress-strain curves to zero plastic strains. i.e. to the intersection of g = EC and 0 = kc:. Figure 3 shows that ~+=e is an approximately linear function of j.,Tpi” with an intercept close to the origin. The equation of the line of best fit passing through the origin is
-E ,
300
(13)
and since a theoretical estimate for the given by Tanaka and Mori [IO] as
stresses is
(1-I)
the unrelaxed strains may also be calculated with f = 0.17, the average of the volume fractions for steels A and B. The values of Aoi and E; are shown in Fig. 4. It is observed that Ao’ increases almost linearly with strain up to ep = 3.57, and tkn remains approximately constant. The unrelaxed strains, E;. impose tensile stresses on the particles tapart from the hydrostatic stress state in the material) approximately equal to 4GA’ e”por 5 G:X%G~l~. where ri’. an “accommodation” factor, has a value of -0.4 for multiple slip [IO. 151. Tlir mrirrrion of h, with main. According to equation (lo), & consists of a strain independent term. kcpcO,and a strain-hardening term. C, <_ The calculated slope k,,= ,, = 1.07 kgf/mm3 I, (equation 11) compares well with the value of 1.1 kgfmm3 ’ obtained by Ball [16] and Warrington [17] for the Hall-Petch slope at yielding for subgrain boundary strengthen20 *5 16’ : ,” -b-
I2
[
/
6. .--
gL
1
-w
_/---
”
_-V
/
-07
0
-06
:a
-05
w
-04
Q 4-
-03 - 0.2 0,
2
3
4
, 5
6
: 7
3
9
01
cp %
Fig. 4. The variation of the averagy interm! stress. AC’. and the unrelaxed plastic strain dlscontinuiry. E;, with strain.
;\NAND
.A>D
CURLASD:
STR.Ab--HARDENING
ing in cold-worked and recovered iron containing 0.03 WLO, impurities. The difference between the observed value of k, (1.23kgf mm3 ‘) at the lower yield stress. equation (1) and the value of i;, of Ball and Warrinpton (1.1 kgf mm3 ‘) is attributed to the enhanced geometrically -necessary dislocation density generated at the particles at the plastic strains associated with the lower yield stress [I]. Our results indicate that E; % E;. and since e”pas calculated from equation (11) represents an upper limit for the unrelaxed plastic strain discontinuity. it is possible to approximate e’p- cp. and from equations (7. 8, 10 and 11) Eve ma) then write I-AGSY= (GC,- G’l).<,- G(“=,,) = C JZ. (IS) where Aa,, is the fraction of the flow stress arising from the dislocation density stored in the material for the geometrical reason of maintaining continuity at the subgrain boundaries within the two-phase polycrystal. ‘4s shown in Fig. 5, the lines of best fit for steels C and D are
16)
OF SPHEROIDIZED
Fig. 6. The deviation dicted by the Ashby
STEELS
905
at E, 2 5,6”, from the kha.vior premodel for incrcsse in Aa, (steel D).
a deviation from Ashby’s theory. which predicts a continuous increase with strain of the dislocation density stored for geometrical reasons. The total increase in How stress at these larger strains should then be dominated by an increase in Aa'. The “tlouhle-rl" behncior. The strain-hardening increment in stress, AGO.may be obtained from equations (7) and (10). AS, =
ctP
-
(T,~=~ =
AG” +
AGO t
C
(18)
%.
\ f!.p and
The relative magnitudes of the three compnents of 17) AG, are shown in Fig. 7 for specimen number 1 (POB) of Table 2. The strong influence of .AG~ suggests the respectively. The two steels separately obey the rela- following explanation for the “double-f)- behavior tion predicted by equation (I 5) because C is a func- observed in spheroidized POB steels: although plastic relaxation occurs at very low strains in the pretion of the volume fraction. The agreement between macroyield region [ 11, the strain-generated dislocaobserved and predicted behavior is good for strains tion density around the particles increasingI> impedes up to 3.50/o.although the small, but non-zero. interthe operation of the secondary dislocation processes cepts indicate the inaccuracy in estimating Ati and AGO.The behavior at strains greater than 3.50/;, is necessary to reduce the stresses in and around the shown for steel D in Fig. 6. It is seen that the stress particles [lS]. This gives rise to the rapid build-up AG,. although still increasing with strain for each of an interphase stress system during the first few persubgrain size. does so at a decreasing rate until an cent of plastic extension (as observed by WYlson and approximately constant value is reached for the speci- Konan [9]). until a limiting value (SG 100) is attained for the particle-matrix interface stresses. At mens with the larger subgrain sizes. This indicates this level of straining, E; and AG’ do not increase any further (Fig. 4) since the stresses in and around the 1 particles then exceed the local yield (or fracrure) stress --
J 9
Fi==. 5. The stress increment AG,, arising from the geometrical dislocation density storedjg_POB specimens as a function of the parameter t (~~!i,,~) for cp _< 3.5!0.
Fig. 7. Magnitudes of the three components of L?IG~for POB spec. No. I at various strains.
906
ANAND
;\sf) GURL.ASD:
STR.‘JN-HARDENING
OF SPHEROIRIZED
STEELS
level necessary to nucleate any secondary processes required to relax the stresses in the particles. Although the effects of a limiting increase in the density of disl~dtions stored for geometrical reasons cannot be neglected. the attainment of a critical level for the particie and particle-rnat~~ interface stresses seems to be the dominant reason for the transition in the strain-hardening behavior observed for these steels (for the volume fractions and particle size ran,oes of Table 2). increment kit vs the parameter , @$,s;i’&for the PI3 specimens at E, I 3.3”;.
Fig. 5. That stress
The microstructure of PIB specimens is charactetized by a random distribution of hard. equiaxed particles in a polycrystalline matrix tvhose grain size is larger than the i~terparticIe spacing. This microstructural characterization approaches the description of an idealized dis~rsion-strengthened system whose strain-hardening behavior may be described by continuum [lo, 11, lS] and quasi-continuum [7] models similar to those above for POB specimens. The fiow Straussmodel. Assuming the presence of long range internal stresses AGO, and the additivity of the hardening contributions AC’ and A(T, arising from the statistically and geometrically stored dislocation densities p” and f. respectively. the stress increat a given strain E, is given ment AGO(= crep- CJ~,=~) by the equation:
best fit describing the particle hardening behavior up to E‘,= 3.5Y, is
The very small value of the intercept and the value of c* % 1, calculated from the slope of equation (23) (C = 526 k&mm’). agree well with Ashby’s theory. St~~iff-~l~~~~~~lj~~ b~~t~~~orof PIB sp~c~ti7~~~s RE stmins yr-ttntwthan X5>;. Ashby’s theory [fl assumes that the geometri~lly necessary secondac dislocations are sessile, uniformly distributed. and continuously stored without attrition in the specimen. At plastic strains ep > 3.59,&however, AGOdoes not conAGO= Ao’ + A8 + AGO. (19) tinue to increase parabolically with strain: this indiIf particle fracture or interface ~~~tation do not cates that as the strain increases the necessary boundary accommodations do not increase proportionately. occur. Ashby calculates AG, as [73: As dispersion strengthened materials are strained, cell structures of dimensions appro~imate1~ quaI to the mean interparticte spacing may form. n’s11 defined celi structures have been observed at high deformations in cold-worked Cu-Si02 and Al-AIIO~ systems where CI is a constant. Two-phase poIycrystaIline by Lewis and Martin [19] and Goodrich and Ansell materials are usuaily characterized by the placement [20]. respectively. For the present work the formaof a certain volume fraction of the second phase particles on the grain boundaries. Assuming that the tion of cell structures in PIB specimens is shown in dominant part of the dislo~tion density stored for the sequence of photomicro~raphs of Fig. 9. At the geometrical reasons is generated at particles of aver- end of Luders straining (at a strain of ep - 1.3%), the d~sl~ation density in the vicinity of the particles is age diameter iii and volume fraction ji situated within the grain volumes. equation (19) may be tvritten as seen to be higher than in the rest of the crystal. At fF _ 5%. the dislocations tend to arrange themselves into faint cell-walls. and at higher strains of E, -. 1046 AGO= . 131) the cell formation becomes more pronounced. A recent review of the complicated many-&x$ problem The quantities AC” and AC’ cannot be calculated of celf formation by Embury[ZI] concludes that the n priori and may only be obtained by experiment. mechanism by which dislocation cells form is still not Estimating AC: as before. from Morrison’s work [ 141 clearly understood. However. it is clear that formaFig. 7. and with the assumption that the average tion of cell walls containing groups of dislocations internal stress AG’ is approximately the same in both of opposite signs allows for some recoxry. and the POB and PIB specimens. the particle contribution interactions between the various disioutions in the Alla,is given as cell walls do not permit a continued increase in the dislocation dwsitv stored for geometric&t reasons. Therefore, the strain-hardening behavior may deviate from that predicted by the Ashby model as cell strucFigure S shows the data for both steels C and D tures start to deveIop. Based on this strain-generated plotted according to equation (27) above. The line of cell structure. it should be possible to rdate the cell
4NX4D
.\XI GURL_GD:
STR,\IN-HARDEMSG
OF SPHEROfDIZED
STEELS
Fig. 9. Dislocation substructures in PTB specimens at various levels of straining. (a) cp = 1.3”,, (compression). Dislocations are tangled around the cementite particles. (b) E, 2 j”;, (tension). Faint dislowion tvalls are discernible. (cl E, z lo”, compression. Ceil structure bccomrs mow pronounced.
bvhere the incrrass in stress required to continue straining consists of(i) the contributions Au, and Ati which arise from the dislocations stored in the material for geometrical and statistical reasons. respectivety and (ii) the contribution AG' arisin,o predominantly from the unrelaxed plastic strain discontinuity between the matrix and the particles. (b) The stress increment AG, arising from the geometrical requirement of maintaining continuity- of the polycrystal is given as
in the rate of strain-hardening (*‘doublw” behavior) observed in these steels. (d) At strains greater than 3%; the increase in G+, appears to be controlled by the increase in A&, because at these strain levels AG’ remains approximately constant and Ao, increases only slightly. 2. The powstressofPIB rprcitmws (a) The flow stress of PIB specimens at plastic strains up to 3.5:; is given by % = ci+=e +- An, + Ati + AG’.
where the constant C depends on the volume fraction of the cementite particles which affect the strain continuity at the subgrdin boundaries on which they are arrayed. The quantity AG’ is obtainable only from experimental measurement. ’ (c) The quantity AC?.representing the average internal back stress. increases almost linearly- up to a plastic strain of 3.5”,, and remains approximately constant thereafter (Fig. 4). The rapid increase in this back stress is a result of the increasing impediment offered by the strain-generated dislocation density around the particles to the secondary dislocation processes which are necessary to relieve the stresses in and around the particles. The limirin,o value of SG; corresponds to the attainment of critical particle or particlcmatrix interface stresses which are sufhcient to nucleate secondary processes to achieve relaxation of the stresses in the partictes. The attainment of this critical stress level also corr?spcnds to the transition
(b) The stress increment Abp arises from the seametrical dislocation density due to the presence of cementite particles of volume fraction _/i of average particle dia. ri; in a matrix of ferrite whose grain size is larger than the mean free path between zmentite particles. It is given by
with C, z 1. (c)The quantities AG”and AG’ have the same vaIues as for POB specimens and hence the origin of the “double-n” behavior in PIB specimens is al= similar to that in POB specimens. (d) At strains e, > 3.51,. Acs, does not increase parabohcally with strain. Also, recovery processes set in which are characterized by dislocation cell formation. The increase in G+ at these strain levels appears to be controlled by an increase in AC’.
AN.AND
CD
GCRLAND:
STR.UN-H~ARDENING
.-lckrwlvled~rmenis-Thc work WBj supported b; Brown Lnikersit) and the L.S. .Atomic Energ) Commisston (now U.S. Energ! Research and Development hdministration) under contract E( 1I-lt!OSA. REF-ERESCES 1. L. Anand and J. Gurland. .Lfet. Trcrns. 7.4. 191 (1976). 2. L. .Anand and J. Gurland. Mrt. Trans. 6.4. 925 (1975). 3. J. Gurland. Sterroloqv and Qmxirarire ~CJrtdloqn~pl~v. p. 108. ASTM, STP- jO4 (1972). 4. L. Anand. Ph.D. Thesis. Brown Universitv (1973). 5. J. H. Hollomon. ~WIS. AlME 162. 268 (I9ij). 6. R. W. Armstrong. I. Codd. R. ivl. Douthwaite and S. J. Petch. Phil. AJag. 7. 45 (1962). 7. !vI. F. Ashbq. Strmgrhrning .LJrrhotis in Crystals (edited by A. Kelly and R. B. Nicholson). p, 137. Wiley. New York t1971). S. A. W. Thompson. Phil. :Llag. 29. 1125 (1974). 9. D. V. %‘iIson and 1’. A. Konan. Actu .Mrr. 12. 617
I I %-+I.
OF SPHEROIDIZED
STEELS
9119
10. K. Tanaka
xtd T. >Iori. .Acm .Llrr. 18. 33 1 (19T01. L. M. Broicn and IV. M. Stobbs. PM. .IJm. 23. I IYi
11971). I?. C. T. Liu. R. ii’. .Armstrong and J. Gur!and. J I S.J. Ii. I-I. II;. 16. 17. IS.
IS. 1393 (1971L l_. F. Kocks. .tlrt. Trwu. 1. I121 (1970). II’. B. Xlorrison. Trms. .-LS.LJ 59. 32-l (19661. J D. Eshslbb. Pvx. R. Sot. J_orr~/. A24l. 376 I 1957~. C. J. Ball. J.l.S.1. 191. 23’ (1959).
D. H. Warrington. J.i.S.1. 201. 610 (1963). L. hf. Brown and \V. .LI. Stobbs. P/G/. .\I+. 23. 1201 (1371L
19. hI. Lewis and J. Martin. .4cn1 Mer. 11. 1X (19631. 20. R. S. Goodrich Jr. and G. S. Ansell. .&x .Ilut. 12. 1097 (1964,. .LJethotls in C’rprds _‘I. J. D. Embury. Srrrnythtwiq (Edited b> A. Kelly and R. B. Nicholson) p. 331. 1Vils). Sew York I 197I I. 27. C. T. Liu and J. Gurland. Trc~ns. T.LIS-.
and J. GCRL.AND
Division of Engineering. Brown University. Providence. RI. U.S.A.
Abstract-The stmin-hardening behavior of spheroidized steels is examined in terms of continuum and quasi-continuum models based on the requirement that continuit! be ~in~ined at various boundaries in the two-phase pol>crystaIs. For plastic strains up to 3.5’,, the increase in the flow stress appears to by composed of contributions arising from (i) dislocations stored in the material for a) statistical and b) geometrical reasons and (ii) an average back stress predominantly due to the presence of the unrelaxed plastic strain discontinuity between the ferrite marrix and the cementite particles. The latter contribution. which is a large fraction of the total strain-hardening increment. increases rapidly with strain up to a plastic strain of 3.Y0 and remains approximately constant thereafter. The attainment of the maximum value of the back stress corresponds to the observed transition in the overall strain-hardening rate (“double-n” behavior). The c~perimentall\ determined components of the hardenine increment due to the geometrically necessary dislocations agree well with the predictions of Ashb& theorv. At strains greaier than 3.YJ0.the average back stress remains approuimatsly constant and the geometrically necessary dislocation density increases only slight@ with strsin. This indicates a deviation from Ashby’s theory which predicts a continuous increase of this dislocation population with strain. The strain-hardening increments at these strain levels therefore appear to be controlled main]! b> the increase in the statistical dislocation population.
R&m&On examine I’&rouissage d’acie:s i carbone sphCrdidal dsns des mod&s continus et quasicontinus qui reposent sur Is maintien de la continuiti aus divers in&aces des polycristaux biphases. Pour des dSformations plastiques allant jusqu’8 3.Y0. il semble que l’accroissement de la contrainte d’&oulement soit composi de contributions provenant (i) des dislocations smmagasinies dans It? materiau pour des raisons a) statistiques et b) giomttriques. et (ii) d’une contrainte interne due essentieilement B la prksence d’une discontinuiti de la d&formation plastiqus non relaxis entre la matrice ferritique et les particules de cementite. La derniere contribution. qui constitue une part importante de I’augmentation totale de durcissement, augmente rapidement avec la diformation jusqu.8 des defermations plastiques de 3.Y,,. puis reste pratiquement constante. Le moment oit la contrainte interrr atteint sa valeur maximale correspond 6 la transition observee dans la vitesse globale d‘ecrouissage. Les composantes exp&imentales de i’accroissement du durcissement dii aux dislocations geom&tiquement nkcessaires sont en bon accord avec Iss predictions dc la theorie
* ;Vow at Research Laboratory, U.S. Steel Corporation. 84onrotville. PA. U.S.A. 901
ANAND X?;D GURLAND:
902
STRAIN-HARDENING
OF SPHEROID~ED
Table 2. ~iicros~~ctural
KTRODUCI’ION X previous paper [l] considered the combined effects of cementite partides. grain boundaries and subgrain boundaries on the room temperature yielding behavior of spheroidized plain carbon steels. The spheroidized structures were obtained by austenitizing and quenching followed by either tempering at temperatures just under the A, temperature or thermal cycling about the ,-t, temperature. Electron microscopy revealed a subgrain-connects cementite particle distribution in the steeis which were spheroid~ed by quenching and tempering [l, 21, and a subgrain-free cementite particle distribution in large ferrite grains in the steels which were spheroidized by the quenchand-cycle procedure. The two different microstructural ty-pes were designated POB (particles on boundaries) and PIB (particles inside boundaries) respectively. In the POB specimens the subgrain size, E.t,,, stabilized by the particles. was found to control the lower yield stress via the relation G,. =
9.5
+
1.3%.$
’
(kgf/mm’)
11.3
+
parameters
POB specimens sp. No.
k.p (lrm)
Steel
f
C D” D
0.33 0.57 0.72 0.53 0.90 1.25
1.19 2.03 2.57 1.53 2.57 3.59
0.6 1 1.31
3 4 :
0.16 0.16 0.16 0.19 0.19
7
D
0.19
1.32
3.SO
2.01
(l)
(2)
(&
1
2
(&
k$ 2.14 t.59
PIB specimens
Sp. No.
Steel
I;
1 2 3 4
C C :
0.12 0.0s 0.15 0.19
0.26 0.36 0.54 0.92
0.52 1.69 1.2: 2.33
1.s-k 3.01 3.49 Il.41
:,
D
0.16 0.14
0.98 0.77
2.57 2.97
10.96 11.81
(1)
The looser yield stress of the PIB specimens was found to be predominantIy controlled by the ferrite grain size. i,. via the Hall-Petch relation Gy =
STEELS
1.87& ’ ” (kgf/mm’).
(2)
In PIB specimens the intraboundary particles at yielding contribute only a small strain-hardening term which increases the value of the friction stress (12.4kgfmm’) over that associated with grain boundary strengthening alone (8.8 f 0.8 kgf/mm’ [3]). It is the purpose of the present work to report the rest&s of an investigation of the strain-hardening behavior of spheroidized high carbon steels in both the quenched-and-ternary (POB) and quench~-andcycled (PIB) microstntctural conditions. An attempt is made to account for the various contributions to the stress increment of strain hardening on the basis of recent theories of particle and boundary strengthening. EXPERIMENTAL The chemical analyses of the two carbon steels are given in Table I. The heat treatment and metallographic procedures are described elsehwere [ 1,4]. Table 2 lists the volume fractioni the mean spherical particle dia.. 6 the mean free path between the particles of cementite. i,. the subgrain size, & (which
is the mean free path considering both particles and sub-sundries as barriers), for PO3 specimens, and the volume fraction,J, and particle dia.. Jr. of cementite particles lying within ferrite grains of mean linear intercept length i., for PIB specimens. Load-elongation curves in tension were obtained at room temperature with l-in. (2.54 x lo-’ m) gauge length, 0.25in. (4.4 x 10T3 m) dia. specimens on an Instron machine at a cross-head speed of O.O&in/min (8.4 x 10m6m’s). The load-elongation data were converted to true stress, G, true totat strain, E, and plastic strain E+,. where (31 EP = E - G/E. According to the Hollomon equation [5]. a=kE”,
(4)
the relation between ha and In E should be linear over the full range of strains, but, in our case, the data for each POB or PIB specimen may be approximated by two straight lines which intersect at about 5% strain. An example of this “double n” behavior is shown in Fig. I for PIB specimen No. 4 of Table 2. The calculated values of the variables and constants 42 4.1 r
b
Table 1. Composition of carbon steels Steel
C
c D
1.05 1.24
Composition* Mn Si 0.34 0.17
0.24 0.16
(wt.%) Al
P
S
0.008 0.009
0.012 0.012
0.011 0.013
* Chemical anal>-sis by Walter M. Saunders, Inc., Provi__
cience.
Kl.
I
-303
-343
-1 is
In*
Fig. 1. Double-n stmi~-hardening
behavior.
-223
n
of equation
(4 for both true total strains and plastic strains are reported in C-l].
-0
c
,.-
1
j-zE,_
”
s ‘M3C’_
‘?O
RESC-LTS .45-D DISCLSSIOS
r\rmstrong et nl. [6] have shown that at temperatures where grain boundary sliding and diffusion do not play a major role. the Aow stress. G,~. of a singlephase polycrystatline metal at a given plastic strain, fp. depends on the grain size. & through the rshtionship Gcp=
Gc,sp
+
ken i-9 ’ :,
f3,
where G~.~ I and k,, are experimental constants. In the case of spheroidized steels with subgrain boundaries pinned by cementite particles (POB). the yield strength is governed by the subgrain size (equation 1) [l]. and it may be assumed that the subgain boundaries behave in a manner analogous to grain boundaries also during strain-hardening. and
unfortunately, it is dif%cult to produce a large range of & for any one given steeI bq simple quenching and tempering. and an accurate determination of Gor, and kcP is not always possible. To obtain approximate estimates of these parameters, the data for steels C and D (which have volume fractions f equal to 0.16 and 0.19, respectively. i.e. values which are reasonably close to each other) were plotted together according to equation (6). and the dependence of the flow stress on the subgrain size is shown in Fig. 2. The values of G~,~, and kcP as determined by least squares anaiysis are listed in Table 3. T~~~~~~ stress ~10~~~.The flow stress at a strain E, is the result of strain induced hardening effects added to the initial yield strength at the very beginning of plastic deformation. The latter is given by equation (6) at zero plastic strain: G<,=()= 00,#,=0 -4”kQZO&’ 2.
(7)
The strain dependent hardening increment is accounted for by the dislocation density and by iongrange internal stresses opposing dislocation motion. The total dislocation density, at a particular value of the strain. is the sum of the statisticaily stored dislocation density. p’. and a geometrically necessary dislocation density. py [7]. The former results from chance encounters and mutual trapping of dislocations within the crystal. the latter is due to the requirement that continuity be maintained at internal boundaries of the polycrystal. The contribution Ao’ of the statistical dislocation density to the flow stress is assumed to be linearly superposed on A@, the con*Ths combined How stress increment should actually be the square root of the sum of the squares of Ati and brig. The above assumption of liner superposition makes The subsequent analysis more tractable.
x.2.P -i
,mm-+
Fig. 2. The relationship between the tiow stritss 0;” at two piastic strains, E,. and the subgram size. f.(.,
tribution of the geometrically necessary dislocation density [S].* Wilson and Monan [9] showed that. in a highcarbon steel spheroidized by quenching and tempering. unloading after tensile deformation left a residua1 stress system in the specimens such that compressive stresses in the ferrite were balanced by tensile stresses in the cementite. The interphase stress system increases rapidly during the first few percent of plastic extension and remains almost constant thereafter. The origin of these internal stresses has recent{> been considered theoretically on the basis of continuum models [lo. 1I]. and their presence is attributed to the unrelaxed plastic strain discontinuity M\veen the matrix and the elastically deforming parricles. The lesser contribution of the single phase pol!crystolline iron matrix to the internal back stresses may be neglected in steels with ;I large volume fraction of cementite. In spheroidized high carbon steels, therefore. the total strain in the matrix is subdivjded into two components. E; and EL, where l “pis the unrelasrd plastic strain discontinuity due to the particles and EL is the portion of the strain which has been relaxed by secondary dislocation processes [7. 11, 171. The unrelaxed strain gives rise to a hardening term. Aa’. which represents an average internal stress opposing Table 5. Calculated values of the constants c,~.<, and ke, in the Hall-Petch equation zSe = bO.cp- kS, ;.,i ’ for the POB specimens
904
ANAND
AND
GCRLAND:
S-IX~IN-HARDENING OF SPHEROIDIZED STEELS
dislocation motion. The relaxed strain portion determines the geometrically-necessary dislocation density and the resulting flow stress increment is. after Ashby [7]. Aa,, = (C, &-;
I,
(S)
where it is assumed (1) that the particle-pinned subgrain boundaries in the POB specimens are analogous to grain boundaries in their effect on generating a dislocation density [9], (2) that the particles uniformly affect the strain continuity at the subgrain boundaries (the constant C is then a function of the volume fraction of the second phase), and (3) that the geometrically necessary dislocations so generated are uniformly distributed within the volumes of the small subgrains under consideration. In summary, the flow stress of POB steels is given by: 0cp = 6cp=,, + do, + A& + Ao’
(9)
or in terms of equation (6):
50
;
40
E \
fy
I
SYMBOL
STEEL
c
.
0
.
+ 400
1
m x 30
-
.
ucs’o = l.O7i.,T,L2.
(11)
The zero value of the intercept is somewhat difficult to rationalize. Liu et al. [12] showed that the lattice friction stress in steels at the onset of plastic flow, as estimated by two alternative double-extrapolation techniques, is uO.~p=O= 3.66 kgf/mm’. Part of the discrepancy may be attributed to the limited data of Fig. 3. In view of the uncertainty, this quantity will be neglected with the realization that the following arguments are not affected significantly by the precise value of G~,~~=~. From equation (10) we may therefore write o,,~, = Aa’ + Agi.
(12)
There are (as yet) no simple geometrical arguments which can be used to calculate the fraction Ao’ of CJ~,~,arising from the statistical dislocation density p’, although it may be obtained from experimental measurements. The ideal statistical hardening curve is a Taylor-corrected stress-strain curve for a single crystal deforming by multiple slip or polyslip [13]. The strain-hardening behavior of a polycrystal with grain size much larger than the average slip distance of a single crystal, however. does approach the Taylor-corrected single-crystal strain-hardening behavior [S]. Therefore the data for a large-grained iron specimen (28Omm grain size. 0.02 wt.% impurity)
z
0
h bW
20
.
.
200
IO 20
I
26
1100
I
I
I
24
32
36
4:
44
-i ’ A:., ) mm-* Fig. 3. The How stress at zero plastic strain. G+=~. as a function of the subgrain size. il.;.
from the work of iMorrison [14] is used here to estimate Ao’ as a function of ep. by means of equation (4). The calculated values of A8 are plotted in Fig. 7. The average internal stresses Ao’ may now be calculated from AC+= r~,,~, - Ad
71~~r@cts of strain on CJ,,(~. The values of a,,= ,, were calculated from the extrapolation of the initial portion of the homogeneous part of the stress-strain curves to zero plastic strains. i.e. to the intersection of g = EC and 0 = kc:. Figure 3 shows that ~+=e is an approximately linear function of j.,Tpi” with an intercept close to the origin. The equation of the line of best fit passing through the origin is
-E ,
300
(13)
and since a theoretical estimate for the given by Tanaka and Mori [IO] as
stresses is
(1-I)
the unrelaxed strains may also be calculated with f = 0.17, the average of the volume fractions for steels A and B. The values of Aoi and E; are shown in Fig. 4. It is observed that Ao’ increases almost linearly with strain up to ep = 3.57, and tkn remains approximately constant. The unrelaxed strains, E;. impose tensile stresses on the particles tapart from the hydrostatic stress state in the material) approximately equal to 4GA’ e”por 5 G:X%G~l~. where ri’. an “accommodation” factor, has a value of -0.4 for multiple slip [IO. 151. Tlir mrirrrion of h, with main. According to equation (lo), & consists of a strain independent term. kcpcO,and a strain-hardening term. C, <_ The calculated slope k,,= ,, = 1.07 kgf/mm3 I, (equation 11) compares well with the value of 1.1 kgfmm3 ’ obtained by Ball [16] and Warrington [17] for the Hall-Petch slope at yielding for subgrain boundary strengthen20 *5 16’ : ,” -b-
I2
[
/
6. .--
gL
1
-w
_/---
”
_-V
/
-07
0
-06
:a
-05
w
-04
Q 4-
-03 - 0.2 0,
2
3
4
, 5
6
: 7
3
9
01
cp %
Fig. 4. The variation of the averagy interm! stress. AC’. and the unrelaxed plastic strain dlscontinuiry. E;, with strain.
;\NAND
.A>D
CURLASD:
STR.Ab--HARDENING
ing in cold-worked and recovered iron containing 0.03 WLO, impurities. The difference between the observed value of k, (1.23kgf mm3 ‘) at the lower yield stress. equation (1) and the value of i;, of Ball and Warrinpton (1.1 kgf mm3 ‘) is attributed to the enhanced geometrically -necessary dislocation density generated at the particles at the plastic strains associated with the lower yield stress [I]. Our results indicate that E; % E;. and since e”pas calculated from equation (11) represents an upper limit for the unrelaxed plastic strain discontinuity. it is possible to approximate e’p- cp. and from equations (7. 8, 10 and 11) Eve ma) then write I-AGSY= (GC,- G’l).<,- G(“=,,) = C JZ. (IS) where Aa,, is the fraction of the flow stress arising from the dislocation density stored in the material for the geometrical reason of maintaining continuity at the subgrain boundaries within the two-phase polycrystal. ‘4s shown in Fig. 5, the lines of best fit for steels C and D are
16)
OF SPHEROIDIZED
Fig. 6. The deviation dicted by the Ashby
STEELS
905
at E, 2 5,6”, from the kha.vior premodel for incrcsse in Aa, (steel D).
a deviation from Ashby’s theory. which predicts a continuous increase with strain of the dislocation density stored for geometrical reasons. The total increase in How stress at these larger strains should then be dominated by an increase in Aa'. The “tlouhle-rl" behncior. The strain-hardening increment in stress, AGO.may be obtained from equations (7) and (10). AS, =
ctP
-
(T,~=~ =
AG” +
AGO t
C
(18)
%.
\ f!.p and
The relative magnitudes of the three compnents of 17) AG, are shown in Fig. 7 for specimen number 1 (POB) of Table 2. The strong influence of .AG~ suggests the respectively. The two steels separately obey the rela- following explanation for the “double-f)- behavior tion predicted by equation (I 5) because C is a func- observed in spheroidized POB steels: although plastic relaxation occurs at very low strains in the pretion of the volume fraction. The agreement between macroyield region [ 11, the strain-generated dislocaobserved and predicted behavior is good for strains tion density around the particles increasingI> impedes up to 3.50/o.although the small, but non-zero. interthe operation of the secondary dislocation processes cepts indicate the inaccuracy in estimating Ati and AGO.The behavior at strains greater than 3.50/;, is necessary to reduce the stresses in and around the shown for steel D in Fig. 6. It is seen that the stress particles [lS]. This gives rise to the rapid build-up AG,. although still increasing with strain for each of an interphase stress system during the first few persubgrain size. does so at a decreasing rate until an cent of plastic extension (as observed by WYlson and approximately constant value is reached for the speci- Konan [9]). until a limiting value (SG 100) is attained for the particle-matrix interface stresses. At mens with the larger subgrain sizes. This indicates this level of straining, E; and AG’ do not increase any further (Fig. 4) since the stresses in and around the 1 particles then exceed the local yield (or fracrure) stress --
J 9
Fi==. 5. The stress increment AG,, arising from the geometrical dislocation density storedjg_POB specimens as a function of the parameter t (~~!i,,~) for cp _< 3.5!0.
Fig. 7. Magnitudes of the three components of L?IG~for POB spec. No. I at various strains.
906
ANAND
;\sf) GURL.ASD:
STR.‘JN-HARDENING
OF SPHEROIRIZED
STEELS
level necessary to nucleate any secondary processes required to relax the stresses in the particles. Although the effects of a limiting increase in the density of disl~dtions stored for geometrical reasons cannot be neglected. the attainment of a critical level for the particie and particle-rnat~~ interface stresses seems to be the dominant reason for the transition in the strain-hardening behavior observed for these steels (for the volume fractions and particle size ran,oes of Table 2). increment kit vs the parameter , @$,s;i’&for the PI3 specimens at E, I 3.3”;.
Fig. 5. That stress
The microstructure of PIB specimens is charactetized by a random distribution of hard. equiaxed particles in a polycrystalline matrix tvhose grain size is larger than the i~terparticIe spacing. This microstructural characterization approaches the description of an idealized dis~rsion-strengthened system whose strain-hardening behavior may be described by continuum [lo, 11, lS] and quasi-continuum [7] models similar to those above for POB specimens. The fiow Straussmodel. Assuming the presence of long range internal stresses AGO, and the additivity of the hardening contributions AC’ and A(T, arising from the statistically and geometrically stored dislocation densities p” and f. respectively. the stress increat a given strain E, is given ment AGO(= crep- CJ~,=~) by the equation:
best fit describing the particle hardening behavior up to E‘,= 3.5Y, is
The very small value of the intercept and the value of c* % 1, calculated from the slope of equation (23) (C = 526 k&mm’). agree well with Ashby’s theory. St~~iff-~l~~~~~~lj~~ b~~t~~~orof PIB sp~c~ti7~~~s RE stmins yr-ttntwthan X5>;. Ashby’s theory [fl assumes that the geometri~lly necessary secondac dislocations are sessile, uniformly distributed. and continuously stored without attrition in the specimen. At plastic strains ep > 3.59,&however, AGOdoes not conAGO= Ao’ + A8 + AGO. (19) tinue to increase parabolically with strain: this indiIf particle fracture or interface ~~~tation do not cates that as the strain increases the necessary boundary accommodations do not increase proportionately. occur. Ashby calculates AG, as [73: As dispersion strengthened materials are strained, cell structures of dimensions appro~imate1~ quaI to the mean interparticte spacing may form. n’s11 defined celi structures have been observed at high deformations in cold-worked Cu-Si02 and Al-AIIO~ systems where CI is a constant. Two-phase poIycrystaIline by Lewis and Martin [19] and Goodrich and Ansell materials are usuaily characterized by the placement [20]. respectively. For the present work the formaof a certain volume fraction of the second phase particles on the grain boundaries. Assuming that the tion of cell structures in PIB specimens is shown in dominant part of the dislo~tion density stored for the sequence of photomicro~raphs of Fig. 9. At the geometrical reasons is generated at particles of aver- end of Luders straining (at a strain of ep - 1.3%), the d~sl~ation density in the vicinity of the particles is age diameter iii and volume fraction ji situated within the grain volumes. equation (19) may be tvritten as seen to be higher than in the rest of the crystal. At fF _ 5%. the dislocations tend to arrange themselves into faint cell-walls. and at higher strains of E, -. 1046 AGO= . 131) the cell formation becomes more pronounced. A recent review of the complicated many-&x$ problem The quantities AC” and AC’ cannot be calculated of celf formation by Embury[ZI] concludes that the n priori and may only be obtained by experiment. mechanism by which dislocation cells form is still not Estimating AC: as before. from Morrison’s work [ 141 clearly understood. However. it is clear that formaFig. 7. and with the assumption that the average tion of cell walls containing groups of dislocations internal stress AG’ is approximately the same in both of opposite signs allows for some recoxry. and the POB and PIB specimens. the particle contribution interactions between the various disioutions in the Alla,is given as cell walls do not permit a continued increase in the dislocation dwsitv stored for geometric&t reasons. Therefore, the strain-hardening behavior may deviate from that predicted by the Ashby model as cell strucFigure S shows the data for both steels C and D tures start to deveIop. Based on this strain-generated plotted according to equation (27) above. The line of cell structure. it should be possible to rdate the cell
4NX4D
.\XI GURL_GD:
STR,\IN-HARDEMSG
OF SPHEROfDIZED
STEELS
Fig. 9. Dislocation substructures in PTB specimens at various levels of straining. (a) cp = 1.3”,, (compression). Dislocations are tangled around the cementite particles. (b) E, 2 j”;, (tension). Faint dislowion tvalls are discernible. (cl E, z lo”, compression. Ceil structure bccomrs mow pronounced.
bvhere the incrrass in stress required to continue straining consists of(i) the contributions Au, and Ati which arise from the dislocations stored in the material for geometrical and statistical reasons. respectivety and (ii) the contribution AG' arisin,o predominantly from the unrelaxed plastic strain discontinuity between the matrix and the particles. (b) The stress increment AG, arising from the geometrical requirement of maintaining continuity- of the polycrystal is given as
in the rate of strain-hardening (*‘doublw” behavior) observed in these steels. (d) At strains greater than 3%; the increase in G+, appears to be controlled by the increase in A&, because at these strain levels AG’ remains approximately constant and Ao, increases only slightly. 2. The powstressofPIB rprcitmws (a) The flow stress of PIB specimens at plastic strains up to 3.5:; is given by % = ci+=e +- An, + Ati + AG’.
where the constant C depends on the volume fraction of the cementite particles which affect the strain continuity at the subgrdin boundaries on which they are arrayed. The quantity AG’ is obtainable only from experimental measurement. ’ (c) The quantity AC?.representing the average internal back stress. increases almost linearly- up to a plastic strain of 3.5”,, and remains approximately constant thereafter (Fig. 4). The rapid increase in this back stress is a result of the increasing impediment offered by the strain-generated dislocation density around the particles to the secondary dislocation processes which are necessary to relieve the stresses in and around the particles. The limirin,o value of SG; corresponds to the attainment of critical particle or particlcmatrix interface stresses which are sufhcient to nucleate secondary processes to achieve relaxation of the stresses in the partictes. The attainment of this critical stress level also corr?spcnds to the transition
(b) The stress increment Abp arises from the seametrical dislocation density due to the presence of cementite particles of volume fraction _/i of average particle dia. ri; in a matrix of ferrite whose grain size is larger than the mean free path between zmentite particles. It is given by
with C, z 1. (c)The quantities AG”and AG’ have the same vaIues as for POB specimens and hence the origin of the “double-n” behavior in PIB specimens is al= similar to that in POB specimens. (d) At strains e, > 3.51,. Acs, does not increase parabohcally with strain. Also, recovery processes set in which are characterized by dislocation cell formation. The increase in G+ at these strain levels appears to be controlled by an increase in AC’.
AN.AND
CD
GCRLAND:
STR.UN-H~ARDENING
.-lckrwlvled~rmenis-Thc work WBj supported b; Brown Lnikersit) and the L.S. .Atomic Energ) Commisston (now U.S. Energ! Research and Development hdministration) under contract E( 1I-lt!OSA. REF-ERESCES 1. L. Anand and J. Gurland. .Lfet. Trcrns. 7.4. 191 (1976). 2. L. .Anand and J. Gurland. Mrt. Trans. 6.4. 925 (1975). 3. J. Gurland. Sterroloqv and Qmxirarire ~CJrtdloqn~pl~v. p. 108. ASTM, STP- jO4 (1972). 4. L. Anand. Ph.D. Thesis. Brown Universitv (1973). 5. J. H. Hollomon. ~WIS. AlME 162. 268 (I9ij). 6. R. W. Armstrong. I. Codd. R. ivl. Douthwaite and S. J. Petch. Phil. AJag. 7. 45 (1962). 7. !vI. F. Ashbq. Strmgrhrning .LJrrhotis in Crystals (edited by A. Kelly and R. B. Nicholson). p, 137. Wiley. New York t1971). S. A. W. Thompson. Phil. :Llag. 29. 1125 (1974). 9. D. V. %‘iIson and 1’. A. Konan. Actu .Mrr. 12. 617
I I %-+I.
OF SPHEROIDIZED
STEELS
9119
10. K. Tanaka
xtd T. >Iori. .Acm .Llrr. 18. 33 1 (19T01. L. M. Broicn and IV. M. Stobbs. PM. .IJm. 23. I IYi
11971). I?. C. T. Liu. R. ii’. .Armstrong and J. Gur!and. J I S.J. Ii. I-I. II;. 16. 17. IS.
IS. 1393 (1971L l_. F. Kocks. .tlrt. Trwu. 1. I121 (1970). II’. B. Xlorrison. Trms. .-LS.LJ 59. 32-l (19661. J D. Eshslbb. Pvx. R. Sot. J_orr~/. A24l. 376 I 1957~. C. J. Ball. J.l.S.1. 191. 23’ (1959).
D. H. Warrington. J.i.S.1. 201. 610 (1963). L. hf. Brown and \V. .LI. Stobbs. P/G/. .\I+. 23. 1201 (1371L
19. hI. Lewis and J. Martin. .4cn1 Mer. 11. 1X (19631. 20. R. S. Goodrich Jr. and G. S. Ansell. .&x .Ilut. 12. 1097 (1964,. .LJethotls in C’rprds _‘I. J. D. Embury. Srrrnythtwiq (Edited b> A. Kelly and R. B. Nicholson) p. 331. 1Vils). Sew York I 197I I. 27. C. T. Liu and J. Gurland. Trc~ns. T.LIS-.