The rate of dislocation multiplication in polycrystalline iron

Materials Science and Engineering, 11 (1973)247-254 © American Societyfor Metals, Metals Park, Ohio, and ElsevierSequoia S.A., Lausanne - Printed in the Netherlands

247

The Rate of Dislocation Multiplication in Polycrystalline Iron W. ROBERTS*, S. K A R L S S O N and Y. B E R G S T R O M

SwedishInstitutefor Metal Research, Stockholm (Sweden) (ReceivedJune 23, 1972)

Summary** A recently presented model for the plastic deformation of ct-iron is used in elucidating those factors which influence the rate of dislocation multiplication during straining of pure iron polycrystals. In particular the effect of grown-in dislocations on the multiplication rate is investigated and the results can be satisfactorily explained in terms of trapping and storage of mobile dislocations. INTRODUCTION The process of dislocation multiplication and its relation to strain hardening are of fundamental importance in the understanding of the mechanical behaviour of metals. It is now well established both theoretically 1- a and experimentally 4'5 that a rather simple relationship exists between flow stress (tr) and dislocation density (p) for a variety of deformation conditions, i.e. tr = trio + 0~Gb p~,

(1)

where G is the shear modulus, b is the magnitude of the dislocation Burgers vector and a and trio are constants. Accepting the validity of eqn. (1), the basic problem in predicting strain hardening is to establish the strain dependence of p. Now irrespective of the detailed mechanism o fmultiplication, the average distance moved by a dislocation at any moment during deformation must be important in determining the p-e (strain) relationship for a constant rate of straining. The purpose of this paper is to establish and discuss those factors which determine this dislocation mean free path in pure polycrystalline iron. Whilst the detailed measurements should be considered specifically only in * Present address: Department of Metallurgy, University of Strathclyde, Glasgow C1, Scotland. ** R6sum6en fran~ais/~la fin de l'article. Deutsche Zusammenfassungam SchluBdes Artikels.

relation to the materials tested, the general conclusions are thought to apply to other metals and alloys in addition to iron. THEORETICAL CONSIDERATIONS

The relationship between the mean free path and the rate of strain hardening An assessment of the mean free path of mobile dislocations during plastic deformation of a material is not simple and some assumptions clearly have to be made. In this paper, we will employ a recently published model aimed at describing the stressstrain behaviour of ~-iron 6 ; this approach provides the necessary connection between strain hardening and the mean free path (s). However, it is only valid under conditions where the assumptions used in developing the model are not seriously violated. In formulating an expression for p(e) only the average behaviour of large numbers of dislocations are considered and the following assumptions are made : (a) The total dislocation density has two components--a mobile density (L) and an immobile density (Pi) (b) L is strain independent. This assumption is at least plausible for iron on the basis of strain-rate change experiments 7,a (c) The variation of Pi with e is determined by the creation, the immobilisation, the remobilisation and the annihilation of dislocations. The mathematical development necessitates the introduction of three main parameters, i.e. U'(e), the rate at which mobile dislocations are immobilised or annihilated; f2', the probability for remobilisation or annihilation of immobile dislocations; al~d A, the rate at which mobile dislocations annihilate with other mobile dislocationsc,specimen surfaces, etc. Invoking assumptions (a)-(c) above the rate of accumulation of dislocations may be written 6 dp= de

U' (e) - A - f2'p.

(2a)

248

W. ROBERTS, S. KARLSSON, Y. BERGSTROM

Strain ageing experiments 9'1° in which dislocation locking permits an estimate of the annihilation contribution to U'(e) and O' indicate that annihilation occurs at a negligible rate except in largegrained material. Equation (2a) can then be written dp de

U (e) - f2p

(2b)

where U(e) is the rate of immobilisation only of mobile dislocations and f2' the strain-independent probability for remobilisation only of immobile dislocations. Simple geometry shows that the rate U for % strain is 10-2 v

-

4,bs(

(3)

) '

where ~b is an orientation factor relating shear and tensile quantities (0.5 for polycrystalline iron). For polycrystalline iron the relationship p = constant, e + P0

(4)

is observed experimentally for small e(< 5%), e.9. see Keh and Weissmann 5. Here, Po is the dislocation density in the undeformed specimen. This initial linear variation of p with strain strongly implies (a) that 12 must be small enough for the term f2p (eqn. (2b)) to be negligible at small strains and (b) that the dislocation mean free path is initially strain independent (eqn. (3)). At larger strains ( > 5 %) a dislocation cell structure is developed which changes little in scale as deformation proceeds 5 ; this suggests that s is strain independent for large e. A reasonable supposition for pure polycrystalline iron is therefore that s is independent of e over the entire stressstrain curve. Integration of eqn. (2b) can then proceed directly with U independent of strain, i.e.

P=~U ( l _ e _ m ) + p 0 e _

m

(5)

which with eqn. (1) gives for a(e)

a=aio+ocGb

(1-e-a~) +po e-a~

.

(6)

These equations are known to describe a(e) and p (e) for iron and steel tested under a variety of conditions 6,9,t°. In all cases, 12 is less than 0.08 confirming the compatibility of eqns. (2b) and (4)for small strains. Furthermore, all parameters vary in a manner consistent with the physical interpretation they are given. The only independently variable parameters in eqn. (6) are U and O. Alpha can be measured from

the slope of an experimental a - p ½ plot according to eqn. (1). The friction stress, aio, is not a variable parameter but is obtained simply as a constant difference between the observed stress-strain curve and that calculated with specific values of U, ~ and f2. It should also be pointed out that U and f2 are only independently variable within fairly narrow limits; experimental data inform us that f2 must be fairly small (eqns. (2b) and (4)) while U must (via eqn. (3)) give an acceptable value for the mean free path. Taking s as 1-5 ~m (compatible with cell sizes in deformed iron polycrystals 5) and b = 2.5 x 10-8 cm, eqn. (3) gives U in the range of 1.6--8 x 109 cm- 2 for each % of strain. Application of eqns. (5) and (6) to experimental a(e) and p(e) data for polycrystalline iron 6 gives U = 2.0 x 109 cm-2 and f2 = 0.057 which seem quite acceptable in the light of the above considerations.

Grown-in dislocations and the mean free path The accumulation of immobile dislocations and the concomitant work hardening can arise either randomly during deformation by some trapping process or through the necessity to deform various parts of the specimen in a compatible way. The latter process is important for the deformation of two-phase alloys in which one component is much stronger than the other (see Ashby11). For the "pure" metals investigated here only the former process is important. In particular, dislocations present prior to deformation--the "grown-in" dislocations--are apparently very effective trapping centres and can determine the random storage of dislocations during straining. Reid et al. ~2 have drawn attention to a rather striking and general correlation between the initial slope of the dislocation density-strain relationship for a material and the corresponding density (P0) of grown-in dislocations. Their findings (see Fig. 1) indicate that for Po > 104 cm-Z U = K p0~

(7)

since U = (dp/de)~= o (eqn. (2b)). This in turn implies that the mean free path is related to the average spacing of grown-in dislocations as (eqn. (3)) 1

s =

- -

.

1

aO0~bbK p~'

(8)

where K is apparently a universal constant. In polycrystalline iron, the grown-in dislocations must be a particularly effective impediment to the motion

DISLOCATION MULTIPLICATION IN POLYCRYSTALLINE Fe

249

9 mm 2 cross-section were heat treated to produce a range of grain sizes at temperatures in the ferritic and austenitic ranges. These procedures will be termed annealing and normalising respectively. Tensile tests were carried out in a Zwick Machine at room temperature and a strain rate of 9.7 x 10- ~ s e c - 1.

Thinned specimens of both deformed and undeformed material were examined in a JEOL 1000D electron microscope operating at 1 MV. The following procedures were adopted for the measurement of dislocation densities.

oYOo 103

7

E

u

'F

102

l

lnSb

10

10

10?

J

I

108

109

1=010

10

U0 crr~ 2. per cent -1

Fig. 1. The relationship between U and the square root of the density of grown-in dislocations for a number of different materials. After Reid, Gilbert and Rosenfield 12.

of mobile dislocations because of pinning by interstitial impurities. The object of this investigation was to measure Po for various types of iron treated in different ways a n d then to correlate the results with U-values obtained via the application of eqn. (6) to appropriate stress-strain data. In this way the validity of eqns. (7) and (8) could be assessed for iron in particular.

EXPERIMENTAL PROCEDURE

Two types of iron were studied: (a) Orkla iron of composition (wt. ~o) C Mn Si Co S N O 0.003 < 0.0004 < 0.0004 0.007 0.007 0.002 0.07

and (b) a titanium-stabilised iron containing 0.14 wt. ~o Ti in addition to the above impurities. The presence of titanium ensures a low concentration of dissolved interstitials (< 1 p.p.m. C + N) through formation of Ti(C, N). Flat tensile specimens of

Grown-in dislocation density This was rather low (1-5× 108 cm -2) and its measurement necessitated the use of the surface intersection technique. Several foils from each of two separate specimens were examined for each heat treatment and it was generally possible by careful use of the goniometric stage to obtain composite micrographs from extensive areas in which dislocation images from only one operating reflection were visible. The total area of foil used in each determination of Po was 2-4000/~m2; these large areas are only available by use of high voltage electron microscopy which has the additional advantage of enabling examination of thicker areas which are more representative of bulk material. Figure 2 shows an area typical of those used in the measurement of grown-in dislocation density. Dislocation densities after deformation These measurements were effected by counting the number of intersections with circles of known circumference 1a, 14. The foil thicknesses determined stereoscopically in the areas of interest varied between 3000 and 5000 A and a mean value of 4000 A was used for the density determinations. For e< 3 ~ , this method was supplemented by surface intersection counts yielding similar results. In all cases due allowance was made for invisible dislocations but whenever possible a reflection ensuring visibility over all Burgers vectors was chosen.

RESULTS AND DISCUSSION

Stress-strain curves Some typical true stress-true strain curves representative of the different materials and heat treatments are shown in Fig, 3. We will not con-

250

W. ROBERTS, S. KARLSSON, Y. BERGSTROM

Fig. 2. A typical area used for po-measurements. Orkla iron--annealed 48 h, 815°C (average grain diameter 52 #m).

40 • anneated

0_.

I

14' 5,urnIb Orkta iron ~ ~_.~_._..o_.3Z5p m,

~

40 normalised Orklo iron

45pro

--

106~urn 20

(~E20

,~ = 0.88 o _

dpm

Uo= 2.0.10"cm 2 = 0.050

tD LIJ ¢Y

~o

t

0

~okgf/m~

106 32.5

0,5 2,5

~14,5

KO

20

10

U0 = 2,Tx|O9crn-2 C). 0.050 =

145 45

0.3 1,3

0

1~

30

~4o

2~

3o

40

oe

.

.

.

Ti-stooiliseo =ron

C:

~""

35pm " 26pro 75pm

20

20

-

///

,~ = ~ 8

[

.("l = 0.050

~okgf~n~d

d ~ m 26 13.5

,b

2~

2.5 5.7

30

0

A

an e led

C

normolised Orkla 40/~rn

lb

I

2b

30

TRUE STRAI N %

Fig. 3. Experimental stress-strain data for the various materials and heat treatments together with theoretical points (open circles) calculated according to eqn. (6) (G = 8 x 103 kgf/mm 2 ; b = 2.5 x 10- s cm). The different strain hardening rates arising through different U values are illustrated in (d).

D I S L O C A T I O N M U L T I P L I C A T I O N IN POLYCRYSTALLINE

40

251

Fe TABLE 2

Ti - s t a b i l i s e d

iron

26~Jm

Experimentally measured grown-in dislocation densities Material

Grain size

Po (with std. dev.) ( × l0 s)

(u") ~20

~

Annealed Orkla iron U = (2.0_+ 0.1) x 109 cm -2

P io

u_

= 0.88

0 i i 0 10 20 M 104 SQUARE ROOT OF TOTAL DISLOCATIONDENSITY cm-1 Fig. 4. The relationship between flow stress and square root of total dislocation density for deformed Ti-stabilised iron. The error bars represent four standard deviations. 9

Normalised Orkla iron U = (2.7_+0.1) x 109 cm -2 Ti-stabilised iron U = (2.3-+0.1) x 109 cm -2

27 32.5 40 106

3.2 (-+ 0.2) 3.1 ( + 0.1) 3.3 (-+0.1) 3.5 ( + 0.1)

45

5.4 ( ___0.2)

88

5.3 (_+0.2)

145

5.6 ( + 0.2)

19 26 33

4.3(+0.1) 4.1 (+ 0.1) 4.3 (+ 0.1)

lO . 30

Ti - stobilised

iron

26,urn t/

'~'E u 2o Z

Z 0

perimental

t_)

calculated from

S

_m]0

equation

(5)

U0 = 2,3~109 crn2 .fl = 0.050

f I

10 TRUE STRAIN %

2.=0

Fig. 5. The dislocation density-strain plot corresponding to Fig. 4. The full line represents that calculated from eqn. (5) to give to best fit to the experimental points. The error bars represent four standard deviations. TABLE 1 Dislocation cell sizes in deformed Ti-stabilised iron (grain size 26/~m) True strain

Cell size

(%)

(~m)

4.5 7.1 10.5 16.6

4.9_+1.2 4.1_+0.6 3.4_+0.5 3.6_+0.5

sider the tensile test results in any detail here but will simply report the principal conclusions of a full and detailed analysis published elsewhere 15. Some theoretical points calculated according to eqn. (6) are shown in Fig. 3 together with the values used for the various parameters. The constant ~ was measured directly for the Ti-stabilised iron from the slope of a ~-p~ plot (Fig. 4) ; the value obtained was also used in fitting the curves for the Orkla iron. The dislocation density-strain relationship corresponding to Fig. 4 is shown in Fig. 5 where the theoretical curve is calculated from eqn. (5) with the values of U and f2 used for the appropriate stress-strain curve (Fig. 3). The dislocation density measurements show that while f2 is small, the remobilisation of dislocations is n o t negligible since there is appreciable deviation at large strains from the initially linear dependence. The observed U-values are very reasonable; for the Ti-stabilised iron (U=2.3 × 109 cm -2) the calculated mean free path is 3.5 ~m which corresponds nicely to the dislocation cell size in this material (see Fig. 6 and Table 1). It should be noted that in calculating the theoretical points for Fig. 3, the term po e-at has been taken as Po (Table 2); that is, we assume that grown-in dislocations are locked and have no probability to remobilise 9'1°. In any case the contribution from this term to the total p is negligible for e > 2 %. The three types of iron investigated, i.e. annealed Orkla, normalised Orkla and Ti-stabilised, all show different rates of strain hardening. This is illustrated by the lower right graph in Fig. 3. However, within any one of these three distinct groups,

252

W. ROBERTS, S. KARLSSON, Y. BERGSTROM

Fig. 6. Ti-stabilised iron strained 16 %. The large area studied permits an accurate estimate of the cell size.

the familiar grain size dependence of flow stresses arises through a grain size dependence of trio. The rate of strain hardening (that is U, ~ and t2 in terms of this model) is independent of grain size. The implications of this result are not important for this paper in which the rate of strain hardening is of interest; for a complete discussion see Roberts and BergstrSm is. We will simply state that while this observation seems at variance with direct measurements of the p (e) as a function of grain size (e.g. Keh and Weissmann 5 for iron, Conrad et al. x6 for niobium), it does concur with the constancy of the Hall-Petch kf with strain in iron 17 and with the grain-size dependence of logarithmic stress-strain curves ls'lg. Furthermore, many p(e)-grain size measurements, e.g. see ref. 17, have been carried out on specimens with a grain size: thickness ratio smaller than the minimum necessary for "true" polycrystal behaviour 2° and may be misleading.

30

2.5

;E ~,o z.0 u

1.~

Present work Keh (1962) Hahn et a[.(19631

Stope=( 1.2 -* 0.1 ) • 10Scm-t

1

Grown-in dislocation densities and their relationship to strain hardening The differences in stress-strain behaviour between the three sets of material investigated lie in the U-

A 0

2 3 UO= 1(~s crn z per cer~ ~

4

Fig. 7. The experimentally-determined relationship between U and x/Po. The error bars for P0 correspond to two standard deviations.

DISLOCATION MULTIPLICATION IN POLYCRYSTALLINE Fe

value. In view ofeqn. (7) it must be suspected that this in turn reflects different grown-in dislocation densities in the different materials. The relevant data are presented in Table 2 and plotted in accordance with eqn. (7) in Fig. 7. Data for a-iron from earlier investigators 21,zz are included in the same Figure for comparison and seem to correlate satisfactorily with the present results. The linear relationship between U and P~o confirms the findings of Reid et al. lz and the constant K determined from the slope of the line is (1.2+0.1)x 10Scm -a. The apparent insensitivity of Po to grain size within each group of specimens studied (v]z. annealed and normalised Orkla iron, and Ti-stabilised iron) is unexpected but in good accord with the observed constancy of the work hardening parameters U and f2 at different grain sizes in these materials (see above). The relationship between U and Po means that the former is inversely proportional to the average spacing of the grown-in dislocations. This implies in turn that the grown-in dislocations act as trapping centres for mobile dislocations causing tangling and storage. This hypothesis is strongly supported by the electron micrographs (Figs. 2 and 6) which show that the undeformed structure has two nearly orthogonal sets of substantially straight dislocations; these are presumably responsible for the highly rectangular cell structure after deformation. The diameter of the trapping circle surrounding each grown-in dislocation is readily calculated ; it is 1/N p~, where N is the mean number of "grown-in spacings" moved by a mobile dislocation before immobilisation. Further, since the mean free path is N/p~8, then comparison with eqn. (8) implies N-

10 - 2

dpbK

and taking b = 2.5 × 10- s cm, K = 1.2 × 105 cm- 1 we get N = 6. Thus for the prevailing net stress, the diameter of the trapping circle for grown-in dislocations is apparently only one-sixth of their mean spacing. It is difficult to find a theoretical estimate to compare with this value ; a rough guide might be the critical distance for dipole formation between two non-parallel dislocations of opposite Burgers vectors. This is roughly Gb/2Hza where za is the prevailing net stress. With z~=10 kgf/mm2, this works out to be 3.2 × 10 -6 cm or 6.4× 10 -6 cm for the trapping diameter; taking Po as 4 × 108 cm -2 and N = 6 , the experimental value is 8.3 x 10-6 cm. Such complete agreement must be fortui-

253

tous since the value used for the prevailing net stress is highly speculative. An alternative explanation for the relationship between s and Po is that locked grown-in screw dislocations effect scattering of mobile screw dislocations by cross slip and thereby enhance tangling and immobilisation. In this case N is the number of cross slip events experienced by the mobile dislocation before it stops. Making use of calculations by Li 23, the total distance of cross slip of the moving screw dislocation is NGb/2zio where Zio is the friction stress. With rio = 1.5 kgf/mm2 as typical for this work, the distance cross-slipped before storage is 4 × 10 -4 cm which is reasonable being about the same as the mean free path (cell size). However, the trapping hypothesis is to be preferred because for the case of substantial intergranular friction, all grown-in dislocations, whether edge or screw, will trap effectively. Scattering by cross slip, on the other hand, will only occur with like screw dislocations, i.e., just one quarter of the mobile dislocations will on average be affected by this process; other combinations of character or sign will give trapping or annihilation.

CONCLUSION

A dislocation model has been applied to stressstrain curves from two different types of iron heat treated in various ways. Differences in strain hardening rate between the different materials can be accounted for quite precisely by correlating one of the parameters from the model with the density of grown-in dislocations. Variations in the grain size of each material studied, i.e. annealed and normalised Orkla iron and Ti-stabilised iron, are not associated with similar changes in grown-in dislocation density or work hardening rate. These results for iron can be satisfactorily explained by considering that the grown-in dislocations act as trapping centres for mobile dislocations. The trapping diameter of each grown-in dislocation is about one-sixth of their mean spacing.

ACKNOWLEDGEMENTS

The authors wish to thank Professor Bertil Aronsson, Director of the Swedish Institute for Metal Research, for encouragement and for permission to publish this communication.

254,

W. ROBERTS, S. KARLSSON, Y. BERGSTROM

1 P. B. Hirsch, Phil. MOO., 7 (1962) 67~ 2 N. F. Mott, Trans. AIME, 218 (1960) 962. 3 A. Seeger, Dislocations and Mechanical Properties of Crystals, Wiley, New York, 1956, p. 243. 4 J. Bailey and P. B. Hirsch, Phil. Mag., 5 (1960) 485. 5 A. S. Keh and S. Weissmann, in G. Thomas and J. Washburn (eds.), Electron Microscopy and Strength of Crystals, Interscience, New York, 1963, p. 231. 6 Y. Bergstr6m, Mater. Sci. Eng., 5 (1970) 193. 7 J. T. Michalak, Acta Met., 13 (1965) 213. 8 J. C. M. Li, Can. J. Phys., 45 (1967) 493. 9 Y. Bergstr6m and W. Roberts, Acta Met., 19 (1971) 815. 10 Y. Bergstr6m, Mater. Sci. Eng., 9 (1972) 101. 11 M. F. Ashby, Phil. Mag., 22 (1970) 399. 12 C. N. Reid, A. Gilbert and A. R. Rosenfield, Phil. Mag., 12

(1965) 409. 13 R. K. Ham, Phil. Mao. , 6 (1961) 1183. 14 J. W. Steeds, Conf. on Electron Microscopy, Cambridge, Inst. Phys., 1963. 15 W. Roberts and Y. Bergstr~Sm, Z. Metallk., 62 (1971) 752. 16 H. Conrad, S. Feuerstein and L. Rice, Mater. Sci. Eng., 2 (1967) 157. 17 R. W. Armstrong, I. Codd, R. M. Douthwaite and N. J. Petch, Phil. Mag., 7 (1962) 45. 18 W. B. Morrison, Trans. Am. Soc. Metals, 59 (1966) 824. 19 B. W. Christ and G. V. Smith, Acta Met., 15 (1967) 809. 20 R. W. Armstrong, J. Mech. Phys. Solids, 9 (1961) 196. 21 G.T. Hahn, C. N. Reid andA. Gilbert, Rept. No. ASD - TDR 63-325, Batelle Memorial Inst., 1963. 22 A. S. Keh, Direct Observation of Imperfections in Crystals, Interscience, New York, 1962, p. 213. 23 J. C. M. Li, d. Appl. Phys., 32 (1961) 593.

Vitesse de multiplication des dislocathgns dans le fer polycristallin

Geschwindigkeit der Versetzunosmultiplikation polykristallinem Eisen

Les auteurs utilisent un mod61e, propos6 r6cemment pour la d6formation plastique du fer 0t, en vue de d6terminer les facteurs qui influencent la vitesse de multiplication des dislocations lors de la d6formation des polycristaux de fer pur. Ils &udient en particulier le r61e des dislocations fig6es pr6existantes dans le m6canisme de la multiplication. Les r6sultats peuvent &re expliqu6s de mani6re satisfaisante par le pi6geage et le stockage de dislocations mobiles.

Anhand eines k/irzlich vorgestellten Modells fiir die plastische Verformung von 0t-Eisen werden die Faktoren erl~iutert, die die Geschwindigkeit der Versetzungsmultiplikation w~ihrend der Zugverformung reiner Eisenvielkristalle beeinflussen. Insbesondere wird der EinfluB der beim Kristallwachstum entstandenen Versetzungen aufdie Multiplikationsgeschwindigkeit untersucht. Die Ergebnisse k6nnen mit dem Einfangen und Festhalten beweglicher Versetzungen befriedigend erkl~irt werden.

REFERENCES

in