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On the nucleation of martensite in iron precipitates
ON THE
NUCLEATION
OF MARTENSITE
K. E. EASTERLINGt
IN IRON
PRECIPITATES*
and G. C. WEATHERLYI
Copper-rich copper-iron alloys containing small defect-free iron precipitates, which in the austenitic condition have practically the same lattice parameter as the surrounding copper matrix, transform to martensite only when the alloy is cold worked. The transformation occurs by the formation of laths with a Kurdjumov-Sachs orientation relationship. Strain energy calculations indicate that the driving force for the transformation in these precipitates, once having introduced the necessary defect, is large enough to accommodate a lath-shaped martensite in agreement with observation. Energy calculations show that the spontaneous transition of an austenite sphere to a martensite ellipsoid is unlikely; furthermore dislocations moving close to precipitates have no chance of triggering off the transformation. The critical step to bring about the transformation is for the precipitate to be cut by the moving matrix dislocation. However, precipitates smaller than about 200 A in diameter do not transform even although they are cut by dislocations and on this basis it is shown that there is a critical size of martensite nucleus which is only a few atoms in thickness. GERMINATION
DE
LA
MARTENSITE
DANS
LES
PRECIPITES
DE
FER
Les alliages cuivre-fer riches en cuivre contenant des petits precipites de fer exempts de defauts et qui, dans les conditions austenitiques, ont pratiquement le meme parametre de reseau que la matrice de cuivre voisine, se transforment en martensite seulement quand l’alliage est travaille a foid. La transformation se manifeste par la formation de rubans ayant une relation d’orientation de Kurdjmov-Sachs. Les calculs de l’energie de deformation montrent que la force directrice pour la transformation de ces precipites, une fois que le defaut necessaire a Bti: introdlit, est suffisamment grande pour produire une martensite en forme de rubans, ceci &ant en accord avecI’observation experimentale. Les calouls d’energie montrent que la transformation spontanee d’une sphere d’austenite en un ellipsoide de martensite est peu probable; de plus, les dislocations se depla$ant pres des precipites n’ont aucune chance de declencher la transformation. Le point critique correspondant a la transformation se produit lorsque le precipite est coupe par la dislocation se deplagant dans la matrice. Cependant, les precipites inferieurs Q 200 A environ ne se transforment pas, m&me s’ils sont coupes par des dislocations, et les auteurs montrent ainsi qu’il existe une taille critique pour l’epaisseur des germes de martensite &gale a quelques atomes. ZUR
MARTENSITKEIMBILDUNG
IN
EISENSUSSCHEIDUNGEN
Kupferreiche Kupfer-Eisen-Legierungen mit kleinen, dsfektfreien Eisenausscheidungen, die in der Austenitform praktisch denselben Gitterprameter wie die umgebende Matrix haben, wandeln sich nur in Martensit urn, wenn die Legierung kaltverformt wird. Die Umwandlung erfolgt durch Bildung von Latten mit einem Kundjumov-Sachs-Orientierungsverhiiltnis. Berechnungen der Verformungsenergie deuten darauf hin, da5 in Ubereinstimmung mit dem Experiment die treibende Kraft zur Umwandlung in diesen Ausscheidungen nach Einbringung der notwendigen Gitterbaufehler gro5 genug ist, urn das lattenformige Martensit zu akkommodieren. Energierechnungen zeigen, da5 der spontane Ubergang von einer austenitischen Kugel su einem martensitischen Ellipsoid unwahrscheinlich ist; au5erdem kijnnen an der Ausscheidung vorbeigleitende Versetzungen die Umandlung nicht auslosen. Der die Umwandlung ausldsende kritische Schritt ist eine die Ausscheidung schneidende Matrixversetzung. Jedoch wandeln sich Aussoheidungen mit einem Durchmesser kleiner als 200 A such dann nicht urn, wenn sie von einer Versetzung geschnitten werden; auf dieser Grundlage wird gezeigt, da5 es eine kritische MartensitkeimgroDe von nur einigen Atomdurchmessern Dicke gibt.
INTRODUCTION
The transformation to martensite of small defectfree precipitates of y-iron in a matrix of copper (which has almost the same lattice parameter as the austenite) only occurs on cold working the alloy. After plastic deformation precipitates are observed to have a banded structure which has been interpreted to exhibit partial transformation, i.e. thin discs or plates of martensite separated by regions of retained austenite.(l) Magnetic measurements on single crystals of the alloy deformed in tension also provide evidence that the transformation is closely related to the slip * Received October 11, 1968. t Department of Metallurgy, Imperial College, London, S.W.7. England. Now at: Department of Physics, Chalmers University of Technology, Gothenburg, Sweden. $ Department of Metallurgy and Materials Science, University of Toronto, Canada. ACTA 3
METALLURGICA,
VOL.
17, JULY
1969
845
behaviour of the crystals.(2) It has been concluded from these experiments that the martensite is nucleated by matrix dislocations as they cut through precipitates during plastic deforn~ation.(1,2) These. results thus offer experimental confirmation of the defect theory of martensite nucleation.(3) The observations of partial transformation in precipitates have recently been questioned on the grounds that the driving force (elastic plus thermodynamic) to transform these small precipitates must be so high that once having introduced the defect the transformation might go spontaneously to completion, the resulting shape change possibly being accommodated elastically by the matrix.t4) It has also been suggested that the elastic strain fields of dislocations moving near the precipitate, e.g. by the Hirsch bypass mechanism,t5) may provide energy sufficient to set the transformation off.t6)
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In this paper we consider these suggestions both in t,he light of some new experimental results and by considering specifically the elastic strain energies and available free energy involved in the transformation of these small precipitates. In this way we can make certain predictions about the shape of martensite formed. EXPERIMENTAL
WORK
An important requirement in this project was to establish both the orientation relationship between the austenite and martensite, and the mo~hology of the martensite. To achieve this it was desirable to obtain electron diffraction patterns containing both f.o.c. and b.c.c. reflections from samples in which the Fe)‘-‘“” transformation had occurred. For this purpose a Cu-1 wt. % Fe alloy single crystal grown directly in the shape of a tensile test specimen(‘) was solution treated and aged at 700°C for 20 hr to give fully coherent y-iron precipitates of about rjOOA mean diameter. The crystal was drawn to failure and several sections were cut from the un-necked part of t8hetest length for electron microscopy and magnetic tests. Magnetic measurements indicated that the transformation was roughly 60-70x complete in sections other than the necked region. Foil preparation by a 33% nitric-methanol electrolyte at a controlled low temperature@) produces wide areaa of transmittable material but precipitates near the edge of the foil tend to be polished away (see e.g. Fig. 5). Recently a new automatic jet polishing techniquetg’ was tried using a chromic-acetic acid solution at room temperature whieh succeeded in pr~ucing clean thin foils without any preferential attack on the iron. Thus in specimens made by the latter technique the volume fraction of iron to copper increased towards the edge of the foil instead of decreasing as with the former method of polishing. On this basis selected area diffraction at the edge of foils exposed strong b.c.c. reflections in addition to those of the matrix. DIFFRACTION
ANALYSIS
Figure l(a) is an electron diffraction pattern from a sample cut parallel to the primary slip plane of the crystal. The f.c.c. structure is close to the (111) orientation as seen from Kikuchi line patterns deeper into the foil. The close similarity between the copper and y-iron lattices (l) which are in the same orientation means that f.c.c. reflections in this system represent both the copper and austenite lattices. The b.c.o. pattern seen super-imposed on the austenite structure is in the (101) orientation as observed from the indexed pattern in Fig. l(b). We see that in the
FIG. l(a). Electron diffraction pattern of a section cut parallel to the primary slip direction in an aged Cn-I y0 Fe alloy Single Crystal drawn in tension to bring about the Fe?-a’ transformation in precipitates.
Fro. l(b). Indsxed pattern of Fig. I(a), illnstrating the (111)~ (lOl)a, Kurdjumov-Sachs orientation relationship.
present case the Kurdjumov-Sachs lationship holds with :
(1111, /I uol),,;
orientation re-
[oii], - /I [iii],,
More exactly, however, we note from Fig. l(b) that: and that
:
[olij, is -3”
from
[iii],,
[ioil, Is-So
from
[ill],
from
[llO],*.
while : Pl),
is ~2’
These measured angles ilt well with those found by Greninger and Troiano(ls) for an Fe-22 % Ni-0.8 % C martensite which gave (ill), is ~1’ from (lol),; (llo),, is --S&O from [ill],,;
(110) is 24’ from [llT].; (211), is 2’ from [llo],;
EASTERLING
et al.:
NUCLEATION
OF
MARTENSITE
IN
Fe
PRECIPITATES
847
the sense of rotation being, of course, also the same as our’s. Note that some of the a’ reflections exhibit a spread of about 44’ [e.g. at “A” in Fig. l(a)] denoting further slight adjustments in orientation between the austenite and martensite lattices. MORPHOLOGY
AND HABIT MARTENSITE
PLANES
OF THE
In the earlier papers it was suggested(la2) that the martensite nucleates as discs with a {llO}, habit. A closer examination of micrographs taken from the earlier stages of tensile deformation of single crystals (see e.g. Fig. 2) shows that at the nucleation stage, the martensite is probably lath rather than plat,e shaped. Unfortunately these particles are very small and contrast is weak, so that their exact shape and possible habit planes have yet to be determined. At a later stage of deformation partially transformed precipitates are observed as seen in Fig. 3, which is a high resolution dark field picture of precipitates using a {211},, type reflection in the (ill), // (IOl),, primary slip plane. The martensite structure is not very clear in these highly magnified pictures but
FIG. 2. Nucleation of thin martensite particles in a Cu-1% Fe alloy single crystal deformed to Stage I in tension. Note that a matrix dislocation appears attached to one end of a martensite particle (at D).
definite lens-shaped martensite can be seen, examples of which are marked “M”. Examples of the banded structure in precipitates similar to that observed earlier in cold-rolled samples(l) are also seen and the interpretation that thin plates or laths of martensite form separated by regions of retained austenite
FIG. 3. High resolution dark field micrograph using a {211}0r reflection in the (111)~ (101)~’ primary slip plane. The significance of the [lTO]y ~[OOl],, direction to martensite growth is emphasised.
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848
METALLURGICA,
holds on the evidence presented here. Indeed, we observe that in no case has a precipitate completely transformed even at this advanced stage of plastic deformation. If the morphology were lath-shaped, this would agree with observations in the bulk ironcopper alloys (e.g. Fig. 6 of Easterling and Miekk-oja’s paper(l) is typical of lath martensite) and in other low alloy iron-based alloys. The morphology of the martensihe is considered further in a later section. A striking crystallographic feature of the dark field pictures is that in the (ill), (lol),, orientation the straight edges of martensite laths lie along the [IIO], [OOl],, direction. This direction has been observed in other lath-type martensites (11) and seems to be an important direction in the martensite nucleation process.c2) MAGNETIC
MEASUREMENTS
In order to extract information on the nucleation stage of the Fe y-a, transformation, we present some results of experiments carried out at an earlier dateo2) on a polycrystalline Cu-2 wt. % Fe alloy. Tensile test specimens were machined from homogenized samples of the alloy. The test specimens were solution treated and then aged at different times at 650°C to vary the mean precipitate size in each one. The specimens were drawn to failure and samples for magnetic moment measurement were spark machined from un-necked parts of the test lengths as well as from undeformed sections of specimens. The results of these tests are shown in Fig. 4. Allowing for the variation about the mean size, the critical size of precipitate which begins to transform appears from these results to be about lOO-zOO A, even although precipitates smaller than these must be cut by dislocations during deformation. A similar result on the critical size of precipitate but using more refined magnetic techniques and single crystals has been obtained by Niinikoski and Forsten.03) The slight drop in relative magnetic moment of the undeformed samples with ageing time, evident in Fig. 4, is probably due to changes in paramagnetism of the s-copper matrix. The sudden increase in tensile strength which occurs at the critical size of precipitate is assumed to occur when dislocations find it harder to cut through the partially transformed precipitates. STRAIN
ENERGY
CONSIDERATIONS
It is assumed that for small precipitates the strains involved in the FeYda’ transformation can be accommodated elastically, at least in the initial stages. Calculations of the strain energies involved in the formation of a plate-shaped martensite nucleus have
VOL.
17,
1969 Ageing
151
time,
I 50
I 100 Mean
min
(at 65O’C)
I I50
precipitate
diameter,
I 300 %
Fra. 4. Magnetic tests to reveal the critical size of precipitate to begin to transform to martensite, and its effect on the tensile strength of a CU-~‘$?~Fe alloy.
already been made by Eshelby,o*) Christiano5J6) and Kaufman.07) In the present case, however, Eshelby’s theory(l*,‘*) would need modification as we are considering the transformation of a region within a transformed region, which involves the solution of a difficult boundary condition at the austenite-dopper interface. In addition, experimental observations suggest that the present transformation involves a lath not a plate shaped nucleus. If the sphere completely transforms to martensite an exact solution is possible simply by adding the strains due to the precipitation of austenite in copper to those due to the transformation of austenite to martensite. We shall then see that the energy of the shear component of the transformation is an order of magnitude greater than the energy of the dilatational component. As a rough approximation this latter term can be ignored and the transformation of a region within the austenite can be assumed to occur in a stress-free matrix. This should not be too bad an approximation as the magnitude of the strain in the austenite precipitates is only 0.Ol.o) If an ellipsoidal volume, V, transforms in an infinite isotropic matrix, the elastic strain energy of the transformation is given by :(l*,i*) E’, = --gVPijIei,T (1) where eijT are the “stress-free” transformation strains, and PiiT are the stresses inside the particle. These are given by the expression: Pifr = A(ec -
eT)dij + 2p(e,jc -
eUT)
(2) where il and p are the Lame constants, and ei3c are the constrained strains inside the particle. The constrained strains are constant and depend only on eijT and the shape of the particle, i.e. e..c=X.. 23
vkl
ekl IT
(3)
EASTERLING
NUCLEATION
et al.:
OF
where Silk6 is constant, values Of ASif*, for various ellipsoidal morphologies can be found from Eshelby’s work.(l*) Equation (I), (2) and (3) can now be combined to express E, in terms of V, eijT and the elastic constants (which for simplicity are considered here to be the same for particle and matrix).
Consider first the transformation of a spherical region from copper to austenite, and then from austenite to martensite. The strains in the first transformation are a pure dilatation, i.e. (eijT)i = fTA&; in the second transformation we assume that they are a plane strain, i.e. the only non-zero components of (eUT)s are (esaT)s = E and (elsT)%= (ealT)a = $12. The total strains in the transformation are (eijT)r + (ei,T)z. These can conveniently be divided into the dilatatio~l components. ell
T-
-
AT -,
3
022
T=--
AT
and the shear components: Te13
-
T-
033
3’
T = (531
-$--I.6
s/s.
The elastic strain energy for the shear and dilatational components can be evaluated separately,* so that: ES = El -j- E2. If Poisson’s ratio is assumed to be N Q, i.e. A frl 2,~) where p is now the shear modulus, the strain energy of the dilatation is : El fi ,uV(+(A~)~ + #AT6 + $t2), and the shear component, is : E, N & ,uVs2 (from equations 1, 2 and 3). The transformation in the iron precipitates has been observed to obey the Kurdjumov-Sachs relationship(see above), and since the precipitates are nearly pure iron with only a litle copper disolved in them(i) the values for (eijT)p are taken for the pure iron transformation ; i.e. [ = 0.05 and s = 0.18, corresponding to a 5 % expansion and a 10” shear. The precipitation of y-iron in copper involves a dilatation of AT = -0.0375.@) Substituting these values into the expressions above, El N lo3 pV, and E, N 8.5 x 10-a pV. These values must be compared with the * This is possible because the Sisr in equation 3 are zero if they couple a shear strain to an extension, so that the dilatation components of e$ give no contribution to the shear compon. ants of e$, tand vice versa.
MARTENSITE
IN
Fe
PRECIPITATES
849
driving force of the transformation. Since the MS temperature is as high as 666”C~1s~and the present tra~formation is made to occur at room tem~rature, the free energy associated with the transformation is high. Following Kaufman and CohencsQ)we estimate it could be as high as 1000 cals/mole. For a volume V of transformation product, this corresponds to ~135 V Gals. The total strain energy involved in the austenite sphere to martensite transformation is ~9.5 x 1O-s PV and substituting in ,U= 8 x 101r dynes/cm2, ES N 180 V cals. Thus even ignoring surface and kinetic energy terms, a spontaneous sphere + ellipsoid transformation appears unlikely, a result which merely emphasises that a sphere is a poor shape for accommodating a plane strain transformation. b. Role of dislocations We shall now consider the effect on the transformation of dislocations moving near to, but not cutting, the precipitate. Ashby(21) has shown that a hard particle that can only deform elastically undergoes a pure elastic shear strain when the matrix suffers a shear displacement due to slip. If the strains inside the particle are uniform, the elastic strain energy in a spherical particle of volume V is: 4Vpp(a + cf2, where ,uais the shear modulus of the particle, and a and c are related to the shear displacements in the matrix, and at the particle-matrix interface respectively. An upper estimate’21) of a + c is given by a+c=--,
(I* E.lP
where G* is the theoretical shear stress.
Note that if (a + c) > a*/,+ the elastic strains are no longer tolerated, and plastic flow begins at the interface. Assuming that CT*= ,430, (a + cf2 = l/900, and taking ,ur = 8 x loll dynes cm-2, the maximum elastic strain energy imposed by dislocations lying outside the precipitate is ~5 cal per unit vol. A similar figure (-4 cal per unit vol) is reached when the effect of the elastic stresses due to the applied tensile load are considered. Thus the combined effect of both these strain energies is still negligible when compared to the driving force of the transformation (~135 Cal) and the strain energy involved in the sphere to martensite transformation (-I80 Cal), and they will not be sufficient to give a spontaneous transformation. The Ashby analysis assumes that the particles are uniformly strained. When the dislocation lies at the interface of a partially coherent precipitate, this will be a poor approximation and it might be more appropriate to consider the displacements close to the
850
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METALLURGICA,
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1969
FIG. 6. Semi-coherent precipitates of y-Fe in an undeformed sample. The interface dislocation loops appear to be punched into the matrix as precipitates are lost in thinning.
dislocation core. A theoretical approach to t,his problem does not appear feasible at the moment, but the experimental evidence again suggests that these large strains alone are insufficient to nucleate martensite. Figure 5 shows semi-coherent iron precipitates with regular dislocation networks visible at the interfaces of the particles; magnetic measurements of these specimens show that the particles are still fully austenitic. c. J~oTphology Experimentally the transformation is observed to occur by the nucleation of small regions inside t,he austenite precipitates (Figs. 2 and 3) ; these are probably laths or needles at this stage (see experimental section above), and the nucleation of martensite of these shapes will now be considered. Only the shear component of this transformation will be considered, for the reasons given above. If an
ellipsoidal region of volume V undergoes a pure shear transformation, the elastic strain energy is given (18) by E, = ~YPJ’(~/~)~,
(4)
where y = 1 - 2X,,,,, X,,,, has been defined in equation 3, and the only non-zero components of the transformation strain are elsT = eslT = s/2. For good accommodation of the product (i.e. low strain energy), y -+ 0, i.e. S,,,, -+ 4. If the product is plate-shaped, i.e. the equation of the transformed region is x2 + y2 --$-t-,=1, and
22
a >cc,
EASTERLING
et al.:
NUCLEATION
OF
so that if the plate is very thin, there is nearly complete accommodation. On the other hand if t.heproduct is needle-shaped i.e. it’s equation is X2 Y2 + -$+-$-=l,
z2
a>c,
S,,r, = 4
and
pVs2
E, = -.
4
This result is almost identical to that obtained for the sphere (see above), which only indicates that a needle is also a poor shape for accommodating a pure shear transformation. Clearly a plate-shaped morphology will always be energetically favoured, but if for certain reasons this shape of martensite cannot form (e.g. due to problems of thickening which for a plate might involve the expansion of several loops of dislocation) there is still sufficient driving force in this system for the martensite to assume some other morphology intermediate between a needle and a plate. This could be the lath-type martensite where a > b > G, which at the nucleation stage could involve the expansion of a single dislocation loop only. Indeed it is possible to show that there are a number of lath morphologies where the strain energy of the transformation (i.e. the driving force per unit volume) is less than 135 cal. In the present case, as the martensite is nucleated by a single dislocation it does not seem logical to give a “classical” analysis of the nucleation event following Kaufman,07) particularly as the dislocation may contribute a significant part of the surface energy term. We will thus assume that approximately 85% of the driving force is expended in accommodating the shear component, with the other 15 % accommodating the dilatation and remaining surface energy. This then gives the condition that &yps2 Q 115
(from equation 4).
Substituting in for p and X, this leads to the conclusion that S,,,, should be > 0.31. The smallest martensite particles that have been seen in the electron microscope, e.g. Fig. 2, have a ~250 A and c N 25 A. If b 2 75 A, then Xi,,, N 0.3, thus this particle could be accommodated. It is therefore concluded that a lath-shaped product does appear feasible. If the initial nucleus forms with a critical a/b/c ratio this could explain why very small precipitates of 200 A or less in diameter cannot transform to martensite. For example if the critical a/c ratio is 10, and the precipitate is 200 A in dia., the maximum possible value of a would be 100 A and c would be 10 A, i.e. the martensite nucleus would only be a few atom planes thick. This may be the critical thickness for a martensite nucleus.
MARTENSITE
IN
Fe PRECIPITATES
851
CONCLUSIONS
1. The martensitic transformation of iron precipitates occurs by the formation of laths with a Kurdjumov-Sachs orientation relationship. The long edges of laths, in the (1 ll), (lol),, primary slip plane, lie predominantly along the [liO], [OOl],, direction which appears to be an important direction in the nucleation process. 2. In no case has a precipitate completely transformed even at advanced stages of deformation of the alloy. This observation is in agreement with strain energy calculations which indicate that a spontaneous austenite sphere to martensite ellipsoid transformation is extremely unlikely to occur. 3. The strain energy from a dislocation moving close by a precipitate contributes a negligible amount to the driving force required for the transformation. The contribution of elastic stresses due to the applied tensile load is likewise insignificant, in agreement with experiment. 4. The critical step to bring about the transformation is evidently for the precipitate to be cut by the moving dislocation. However, there is also a critical size of precipitate (-200 A dia.) below which the transformation cannot be induced. On this basis, energy calculations point to a critical size of martensite nucleus of about 20 A in thickness. 5. In view of the high MS temperature of iron-rich iron-copper alloys, it is estimated that the driving force at room temperature is sufficient to accommodate the lath-shaped martensite observed. On this basis there would appear to be no need to invoke a “multiple” nucleation event with a number of parallel laths forming a mutually accommodating system. ACKNOWLEDGMENT
The authors are grateful to Dr. P. R. Swarm for his comments and to Professor J. G. Ball for laboratory facilities. The work was started while one of us (G. C. W.) was at the Metallurgy Division, A.E.R.E., Harwell and he is grateful to U.K.A.E.A. for the award of a research fellowship and N.R.C. Canada for financial support. K. E. E. acknowledges support in the form of a Science Research Council award. REFERENCES 1. K. E. EASTERLINQand H. M. MIEEK-OJA, Acta Met. 15, 1133 (1967). 2. K. E. EASTERLIN~and P. R. SWANN, Proc. Cmf. Mechan&m of Phase Transformations in Crystalline Solids. Inst. Met& (1968). 3. J. W. CHRISTIAN, Proc. Cmf. Me&a&am of Phuae Transfomtions in Crystalline Sobid8. Inst. Metals (1968). 4. J. W. CHRISTIAN, Proc. Conf. Mechanism of Phase Transfomnations in Cyetalline Solida, discussion to Ref. 2. Inst. Metals (1968). 5. P. B. HIRSCH,J. Inst. Metals 86, 7 (1957).
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852
METALLURGICA,
6. N. E. RYAN, Proc. Con!. Mechanism of Phase Traneformatione in Crystalline Solids, discussion to Ref. 2. Inst. Metals (1968). 7. V. K. LINDROOS and H. M. MIEKK-OJA, J. Inst. Metals 93, 513 (1964-66).
8. R. RHTY, V. LINDROOS, A. SAARINEN, J. FORSTBN and H. M. MIEKK-OJA, J. scient. In&rum. 45, 367 (1966). 9. H. 0. ALI-EL-AWAD, R. ADAMS and P. R. SWASN, to be published. 10. A. B. GRENINGER and A. R. TROIANO, Trans. Am. Inst. Min. metall. Engrs 185, 590 (1949). 11. J. W. CHRISTIAN, The Theory of Transformations in Metals and Alloys. Pergamon (1965).
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1969
12. K. E. EASTERLINQ, Tekn. lis. Thesis, Finland Technical University ( 1965). 13. T. NIINIKOSKI and J. FORST~N, private
(1968).
communication
14. J. D. ESHELBY, Prog. Solid Mech. 2, 89, (1961). 15. J. W. CHRISTIAN,Acta Met. 6, 377 (1968). 16. J. W. CHRISTIAN, Acta Met. 7, 218 (1959). 17. L. KAUFMAN, Acta Met. 7, 216 (1959). 18. J. W. ESHELBY, Proc. R. Sot. A241, 376 (1957). 19. E. R;~s~NEN, Tekn. lis. Thesis, Finland Technical University (1967). 20. L. KAUFMAP; and M. COHEN, Prog. Metal Phys. 7, 165 (1958). 21. M. F. ASHBY, Phil. Mag. 14, 1157 (1966).
NUCLEATION
OF MARTENSITE
K. E. EASTERLINGt
IN IRON
PRECIPITATES*
and G. C. WEATHERLYI
Copper-rich copper-iron alloys containing small defect-free iron precipitates, which in the austenitic condition have practically the same lattice parameter as the surrounding copper matrix, transform to martensite only when the alloy is cold worked. The transformation occurs by the formation of laths with a Kurdjumov-Sachs orientation relationship. Strain energy calculations indicate that the driving force for the transformation in these precipitates, once having introduced the necessary defect, is large enough to accommodate a lath-shaped martensite in agreement with observation. Energy calculations show that the spontaneous transition of an austenite sphere to a martensite ellipsoid is unlikely; furthermore dislocations moving close to precipitates have no chance of triggering off the transformation. The critical step to bring about the transformation is for the precipitate to be cut by the moving matrix dislocation. However, precipitates smaller than about 200 A in diameter do not transform even although they are cut by dislocations and on this basis it is shown that there is a critical size of martensite nucleus which is only a few atoms in thickness. GERMINATION
DE
LA
MARTENSITE
DANS
LES
PRECIPITES
DE
FER
Les alliages cuivre-fer riches en cuivre contenant des petits precipites de fer exempts de defauts et qui, dans les conditions austenitiques, ont pratiquement le meme parametre de reseau que la matrice de cuivre voisine, se transforment en martensite seulement quand l’alliage est travaille a foid. La transformation se manifeste par la formation de rubans ayant une relation d’orientation de Kurdjmov-Sachs. Les calculs de l’energie de deformation montrent que la force directrice pour la transformation de ces precipites, une fois que le defaut necessaire a Bti: introdlit, est suffisamment grande pour produire une martensite en forme de rubans, ceci &ant en accord avecI’observation experimentale. Les calouls d’energie montrent que la transformation spontanee d’une sphere d’austenite en un ellipsoide de martensite est peu probable; de plus, les dislocations se depla$ant pres des precipites n’ont aucune chance de declencher la transformation. Le point critique correspondant a la transformation se produit lorsque le precipite est coupe par la dislocation se deplagant dans la matrice. Cependant, les precipites inferieurs Q 200 A environ ne se transforment pas, m&me s’ils sont coupes par des dislocations, et les auteurs montrent ainsi qu’il existe une taille critique pour l’epaisseur des germes de martensite &gale a quelques atomes. ZUR
MARTENSITKEIMBILDUNG
IN
EISENSUSSCHEIDUNGEN
Kupferreiche Kupfer-Eisen-Legierungen mit kleinen, dsfektfreien Eisenausscheidungen, die in der Austenitform praktisch denselben Gitterprameter wie die umgebende Matrix haben, wandeln sich nur in Martensit urn, wenn die Legierung kaltverformt wird. Die Umwandlung erfolgt durch Bildung von Latten mit einem Kundjumov-Sachs-Orientierungsverhiiltnis. Berechnungen der Verformungsenergie deuten darauf hin, da5 in Ubereinstimmung mit dem Experiment die treibende Kraft zur Umwandlung in diesen Ausscheidungen nach Einbringung der notwendigen Gitterbaufehler gro5 genug ist, urn das lattenformige Martensit zu akkommodieren. Energierechnungen zeigen, da5 der spontane Ubergang von einer austenitischen Kugel su einem martensitischen Ellipsoid unwahrscheinlich ist; au5erdem kijnnen an der Ausscheidung vorbeigleitende Versetzungen die Umandlung nicht auslosen. Der die Umwandlung ausldsende kritische Schritt ist eine die Ausscheidung schneidende Matrixversetzung. Jedoch wandeln sich Aussoheidungen mit einem Durchmesser kleiner als 200 A such dann nicht urn, wenn sie von einer Versetzung geschnitten werden; auf dieser Grundlage wird gezeigt, da5 es eine kritische MartensitkeimgroDe von nur einigen Atomdurchmessern Dicke gibt.
INTRODUCTION
The transformation to martensite of small defectfree precipitates of y-iron in a matrix of copper (which has almost the same lattice parameter as the austenite) only occurs on cold working the alloy. After plastic deformation precipitates are observed to have a banded structure which has been interpreted to exhibit partial transformation, i.e. thin discs or plates of martensite separated by regions of retained austenite.(l) Magnetic measurements on single crystals of the alloy deformed in tension also provide evidence that the transformation is closely related to the slip * Received October 11, 1968. t Department of Metallurgy, Imperial College, London, S.W.7. England. Now at: Department of Physics, Chalmers University of Technology, Gothenburg, Sweden. $ Department of Metallurgy and Materials Science, University of Toronto, Canada. ACTA 3
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behaviour of the crystals.(2) It has been concluded from these experiments that the martensite is nucleated by matrix dislocations as they cut through precipitates during plastic deforn~ation.(1,2) These. results thus offer experimental confirmation of the defect theory of martensite nucleation.(3) The observations of partial transformation in precipitates have recently been questioned on the grounds that the driving force (elastic plus thermodynamic) to transform these small precipitates must be so high that once having introduced the defect the transformation might go spontaneously to completion, the resulting shape change possibly being accommodated elastically by the matrix.t4) It has also been suggested that the elastic strain fields of dislocations moving near the precipitate, e.g. by the Hirsch bypass mechanism,t5) may provide energy sufficient to set the transformation off.t6)
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In this paper we consider these suggestions both in t,he light of some new experimental results and by considering specifically the elastic strain energies and available free energy involved in the transformation of these small precipitates. In this way we can make certain predictions about the shape of martensite formed. EXPERIMENTAL
WORK
An important requirement in this project was to establish both the orientation relationship between the austenite and martensite, and the mo~hology of the martensite. To achieve this it was desirable to obtain electron diffraction patterns containing both f.o.c. and b.c.c. reflections from samples in which the Fe)‘-‘“” transformation had occurred. For this purpose a Cu-1 wt. % Fe alloy single crystal grown directly in the shape of a tensile test specimen(‘) was solution treated and aged at 700°C for 20 hr to give fully coherent y-iron precipitates of about rjOOA mean diameter. The crystal was drawn to failure and several sections were cut from the un-necked part of t8hetest length for electron microscopy and magnetic tests. Magnetic measurements indicated that the transformation was roughly 60-70x complete in sections other than the necked region. Foil preparation by a 33% nitric-methanol electrolyte at a controlled low temperature@) produces wide areaa of transmittable material but precipitates near the edge of the foil tend to be polished away (see e.g. Fig. 5). Recently a new automatic jet polishing techniquetg’ was tried using a chromic-acetic acid solution at room temperature whieh succeeded in pr~ucing clean thin foils without any preferential attack on the iron. Thus in specimens made by the latter technique the volume fraction of iron to copper increased towards the edge of the foil instead of decreasing as with the former method of polishing. On this basis selected area diffraction at the edge of foils exposed strong b.c.c. reflections in addition to those of the matrix. DIFFRACTION
ANALYSIS
Figure l(a) is an electron diffraction pattern from a sample cut parallel to the primary slip plane of the crystal. The f.c.c. structure is close to the (111) orientation as seen from Kikuchi line patterns deeper into the foil. The close similarity between the copper and y-iron lattices (l) which are in the same orientation means that f.c.c. reflections in this system represent both the copper and austenite lattices. The b.c.o. pattern seen super-imposed on the austenite structure is in the (101) orientation as observed from the indexed pattern in Fig. l(b). We see that in the
FIG. l(a). Electron diffraction pattern of a section cut parallel to the primary slip direction in an aged Cn-I y0 Fe alloy Single Crystal drawn in tension to bring about the Fe?-a’ transformation in precipitates.
Fro. l(b). Indsxed pattern of Fig. I(a), illnstrating the (111)~ (lOl)a, Kurdjumov-Sachs orientation relationship.
present case the Kurdjumov-Sachs lationship holds with :
(1111, /I uol),,;
orientation re-
[oii], - /I [iii],,
More exactly, however, we note from Fig. l(b) that: and that
:
[olij, is -3”
from
[iii],,
[ioil, Is-So
from
[ill],
from
[llO],*.
while : Pl),
is ~2’
These measured angles ilt well with those found by Greninger and Troiano(ls) for an Fe-22 % Ni-0.8 % C martensite which gave (ill), is ~1’ from (lol),; (llo),, is --S&O from [ill],,;
(110) is 24’ from [llT].; (211), is 2’ from [llo],;
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NUCLEATION
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the sense of rotation being, of course, also the same as our’s. Note that some of the a’ reflections exhibit a spread of about 44’ [e.g. at “A” in Fig. l(a)] denoting further slight adjustments in orientation between the austenite and martensite lattices. MORPHOLOGY
AND HABIT MARTENSITE
PLANES
OF THE
In the earlier papers it was suggested(la2) that the martensite nucleates as discs with a {llO}, habit. A closer examination of micrographs taken from the earlier stages of tensile deformation of single crystals (see e.g. Fig. 2) shows that at the nucleation stage, the martensite is probably lath rather than plat,e shaped. Unfortunately these particles are very small and contrast is weak, so that their exact shape and possible habit planes have yet to be determined. At a later stage of deformation partially transformed precipitates are observed as seen in Fig. 3, which is a high resolution dark field picture of precipitates using a {211},, type reflection in the (ill), // (IOl),, primary slip plane. The martensite structure is not very clear in these highly magnified pictures but
FIG. 2. Nucleation of thin martensite particles in a Cu-1% Fe alloy single crystal deformed to Stage I in tension. Note that a matrix dislocation appears attached to one end of a martensite particle (at D).
definite lens-shaped martensite can be seen, examples of which are marked “M”. Examples of the banded structure in precipitates similar to that observed earlier in cold-rolled samples(l) are also seen and the interpretation that thin plates or laths of martensite form separated by regions of retained austenite
FIG. 3. High resolution dark field micrograph using a {211}0r reflection in the (111)~ (101)~’ primary slip plane. The significance of the [lTO]y ~[OOl],, direction to martensite growth is emphasised.
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holds on the evidence presented here. Indeed, we observe that in no case has a precipitate completely transformed even at this advanced stage of plastic deformation. If the morphology were lath-shaped, this would agree with observations in the bulk ironcopper alloys (e.g. Fig. 6 of Easterling and Miekk-oja’s paper(l) is typical of lath martensite) and in other low alloy iron-based alloys. The morphology of the martensihe is considered further in a later section. A striking crystallographic feature of the dark field pictures is that in the (ill), (lol),, orientation the straight edges of martensite laths lie along the [IIO], [OOl],, direction. This direction has been observed in other lath-type martensites (11) and seems to be an important direction in the martensite nucleation process.c2) MAGNETIC
MEASUREMENTS
In order to extract information on the nucleation stage of the Fe y-a, transformation, we present some results of experiments carried out at an earlier dateo2) on a polycrystalline Cu-2 wt. % Fe alloy. Tensile test specimens were machined from homogenized samples of the alloy. The test specimens were solution treated and then aged at different times at 650°C to vary the mean precipitate size in each one. The specimens were drawn to failure and samples for magnetic moment measurement were spark machined from un-necked parts of the test lengths as well as from undeformed sections of specimens. The results of these tests are shown in Fig. 4. Allowing for the variation about the mean size, the critical size of precipitate which begins to transform appears from these results to be about lOO-zOO A, even although precipitates smaller than these must be cut by dislocations during deformation. A similar result on the critical size of precipitate but using more refined magnetic techniques and single crystals has been obtained by Niinikoski and Forsten.03) The slight drop in relative magnetic moment of the undeformed samples with ageing time, evident in Fig. 4, is probably due to changes in paramagnetism of the s-copper matrix. The sudden increase in tensile strength which occurs at the critical size of precipitate is assumed to occur when dislocations find it harder to cut through the partially transformed precipitates. STRAIN
ENERGY
CONSIDERATIONS
It is assumed that for small precipitates the strains involved in the FeYda’ transformation can be accommodated elastically, at least in the initial stages. Calculations of the strain energies involved in the formation of a plate-shaped martensite nucleus have
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time,
I 50
I 100 Mean
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(at 65O’C)
I I50
precipitate
diameter,
I 300 %
Fra. 4. Magnetic tests to reveal the critical size of precipitate to begin to transform to martensite, and its effect on the tensile strength of a CU-~‘$?~Fe alloy.
already been made by Eshelby,o*) Christiano5J6) and Kaufman.07) In the present case, however, Eshelby’s theory(l*,‘*) would need modification as we are considering the transformation of a region within a transformed region, which involves the solution of a difficult boundary condition at the austenite-dopper interface. In addition, experimental observations suggest that the present transformation involves a lath not a plate shaped nucleus. If the sphere completely transforms to martensite an exact solution is possible simply by adding the strains due to the precipitation of austenite in copper to those due to the transformation of austenite to martensite. We shall then see that the energy of the shear component of the transformation is an order of magnitude greater than the energy of the dilatational component. As a rough approximation this latter term can be ignored and the transformation of a region within the austenite can be assumed to occur in a stress-free matrix. This should not be too bad an approximation as the magnitude of the strain in the austenite precipitates is only 0.Ol.o) If an ellipsoidal volume, V, transforms in an infinite isotropic matrix, the elastic strain energy of the transformation is given by :(l*,i*) E’, = --gVPijIei,T (1) where eijT are the “stress-free” transformation strains, and PiiT are the stresses inside the particle. These are given by the expression: Pifr = A(ec -
eT)dij + 2p(e,jc -
eUT)
(2) where il and p are the Lame constants, and ei3c are the constrained strains inside the particle. The constrained strains are constant and depend only on eijT and the shape of the particle, i.e. e..c=X.. 23
vkl
ekl IT
(3)
EASTERLING
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et al.:
OF
where Silk6 is constant, values Of ASif*, for various ellipsoidal morphologies can be found from Eshelby’s work.(l*) Equation (I), (2) and (3) can now be combined to express E, in terms of V, eijT and the elastic constants (which for simplicity are considered here to be the same for particle and matrix).
Consider first the transformation of a spherical region from copper to austenite, and then from austenite to martensite. The strains in the first transformation are a pure dilatation, i.e. (eijT)i = fTA&; in the second transformation we assume that they are a plane strain, i.e. the only non-zero components of (eUT)s are (esaT)s = E and (elsT)%= (ealT)a = $12. The total strains in the transformation are (eijT)r + (ei,T)z. These can conveniently be divided into the dilatatio~l components. ell
T-
-
AT -,
3
022
T=--
AT
and the shear components: Te13
-
T-
033
3’
T = (531
-$--I.6
s/s.
The elastic strain energy for the shear and dilatational components can be evaluated separately,* so that: ES = El -j- E2. If Poisson’s ratio is assumed to be N Q, i.e. A frl 2,~) where p is now the shear modulus, the strain energy of the dilatation is : El fi ,uV(+(A~)~ + #AT6 + $t2), and the shear component, is : E, N & ,uVs2 (from equations 1, 2 and 3). The transformation in the iron precipitates has been observed to obey the Kurdjumov-Sachs relationship(see above), and since the precipitates are nearly pure iron with only a litle copper disolved in them(i) the values for (eijT)p are taken for the pure iron transformation ; i.e. [ = 0.05 and s = 0.18, corresponding to a 5 % expansion and a 10” shear. The precipitation of y-iron in copper involves a dilatation of AT = -0.0375.@) Substituting these values into the expressions above, El N lo3 pV, and E, N 8.5 x 10-a pV. These values must be compared with the * This is possible because the Sisr in equation 3 are zero if they couple a shear strain to an extension, so that the dilatation components of e$ give no contribution to the shear compon. ants of e$, tand vice versa.
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driving force of the transformation. Since the MS temperature is as high as 666”C~1s~and the present tra~formation is made to occur at room tem~rature, the free energy associated with the transformation is high. Following Kaufman and CohencsQ)we estimate it could be as high as 1000 cals/mole. For a volume V of transformation product, this corresponds to ~135 V Gals. The total strain energy involved in the austenite sphere to martensite transformation is ~9.5 x 1O-s PV and substituting in ,U= 8 x 101r dynes/cm2, ES N 180 V cals. Thus even ignoring surface and kinetic energy terms, a spontaneous sphere + ellipsoid transformation appears unlikely, a result which merely emphasises that a sphere is a poor shape for accommodating a plane strain transformation. b. Role of dislocations We shall now consider the effect on the transformation of dislocations moving near to, but not cutting, the precipitate. Ashby(21) has shown that a hard particle that can only deform elastically undergoes a pure elastic shear strain when the matrix suffers a shear displacement due to slip. If the strains inside the particle are uniform, the elastic strain energy in a spherical particle of volume V is: 4Vpp(a + cf2, where ,uais the shear modulus of the particle, and a and c are related to the shear displacements in the matrix, and at the particle-matrix interface respectively. An upper estimate’21) of a + c is given by a+c=--,
(I* E.lP
where G* is the theoretical shear stress.
Note that if (a + c) > a*/,+ the elastic strains are no longer tolerated, and plastic flow begins at the interface. Assuming that CT*= ,430, (a + cf2 = l/900, and taking ,ur = 8 x loll dynes cm-2, the maximum elastic strain energy imposed by dislocations lying outside the precipitate is ~5 cal per unit vol. A similar figure (-4 cal per unit vol) is reached when the effect of the elastic stresses due to the applied tensile load are considered. Thus the combined effect of both these strain energies is still negligible when compared to the driving force of the transformation (~135 Cal) and the strain energy involved in the sphere to martensite transformation (-I80 Cal), and they will not be sufficient to give a spontaneous transformation. The Ashby analysis assumes that the particles are uniformly strained. When the dislocation lies at the interface of a partially coherent precipitate, this will be a poor approximation and it might be more appropriate to consider the displacements close to the
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FIG. 6. Semi-coherent precipitates of y-Fe in an undeformed sample. The interface dislocation loops appear to be punched into the matrix as precipitates are lost in thinning.
dislocation core. A theoretical approach to t,his problem does not appear feasible at the moment, but the experimental evidence again suggests that these large strains alone are insufficient to nucleate martensite. Figure 5 shows semi-coherent iron precipitates with regular dislocation networks visible at the interfaces of the particles; magnetic measurements of these specimens show that the particles are still fully austenitic. c. J~oTphology Experimentally the transformation is observed to occur by the nucleation of small regions inside t,he austenite precipitates (Figs. 2 and 3) ; these are probably laths or needles at this stage (see experimental section above), and the nucleation of martensite of these shapes will now be considered. Only the shear component of this transformation will be considered, for the reasons given above. If an
ellipsoidal region of volume V undergoes a pure shear transformation, the elastic strain energy is given (18) by E, = ~YPJ’(~/~)~,
(4)
where y = 1 - 2X,,,,, X,,,, has been defined in equation 3, and the only non-zero components of the transformation strain are elsT = eslT = s/2. For good accommodation of the product (i.e. low strain energy), y -+ 0, i.e. S,,,, -+ 4. If the product is plate-shaped, i.e. the equation of the transformed region is x2 + y2 --$-t-,=1, and
22
a >cc,
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so that if the plate is very thin, there is nearly complete accommodation. On the other hand if t.heproduct is needle-shaped i.e. it’s equation is X2 Y2 + -$+-$-=l,
z2
a>c,
S,,r, = 4
and
pVs2
E, = -.
4
This result is almost identical to that obtained for the sphere (see above), which only indicates that a needle is also a poor shape for accommodating a pure shear transformation. Clearly a plate-shaped morphology will always be energetically favoured, but if for certain reasons this shape of martensite cannot form (e.g. due to problems of thickening which for a plate might involve the expansion of several loops of dislocation) there is still sufficient driving force in this system for the martensite to assume some other morphology intermediate between a needle and a plate. This could be the lath-type martensite where a > b > G, which at the nucleation stage could involve the expansion of a single dislocation loop only. Indeed it is possible to show that there are a number of lath morphologies where the strain energy of the transformation (i.e. the driving force per unit volume) is less than 135 cal. In the present case, as the martensite is nucleated by a single dislocation it does not seem logical to give a “classical” analysis of the nucleation event following Kaufman,07) particularly as the dislocation may contribute a significant part of the surface energy term. We will thus assume that approximately 85% of the driving force is expended in accommodating the shear component, with the other 15 % accommodating the dilatation and remaining surface energy. This then gives the condition that &yps2 Q 115
(from equation 4).
Substituting in for p and X, this leads to the conclusion that S,,,, should be > 0.31. The smallest martensite particles that have been seen in the electron microscope, e.g. Fig. 2, have a ~250 A and c N 25 A. If b 2 75 A, then Xi,,, N 0.3, thus this particle could be accommodated. It is therefore concluded that a lath-shaped product does appear feasible. If the initial nucleus forms with a critical a/b/c ratio this could explain why very small precipitates of 200 A or less in diameter cannot transform to martensite. For example if the critical a/c ratio is 10, and the precipitate is 200 A in dia., the maximum possible value of a would be 100 A and c would be 10 A, i.e. the martensite nucleus would only be a few atom planes thick. This may be the critical thickness for a martensite nucleus.
MARTENSITE
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CONCLUSIONS
1. The martensitic transformation of iron precipitates occurs by the formation of laths with a Kurdjumov-Sachs orientation relationship. The long edges of laths, in the (1 ll), (lol),, primary slip plane, lie predominantly along the [liO], [OOl],, direction which appears to be an important direction in the nucleation process. 2. In no case has a precipitate completely transformed even at advanced stages of deformation of the alloy. This observation is in agreement with strain energy calculations which indicate that a spontaneous austenite sphere to martensite ellipsoid transformation is extremely unlikely to occur. 3. The strain energy from a dislocation moving close by a precipitate contributes a negligible amount to the driving force required for the transformation. The contribution of elastic stresses due to the applied tensile load is likewise insignificant, in agreement with experiment. 4. The critical step to bring about the transformation is evidently for the precipitate to be cut by the moving dislocation. However, there is also a critical size of precipitate (-200 A dia.) below which the transformation cannot be induced. On this basis, energy calculations point to a critical size of martensite nucleus of about 20 A in thickness. 5. In view of the high MS temperature of iron-rich iron-copper alloys, it is estimated that the driving force at room temperature is sufficient to accommodate the lath-shaped martensite observed. On this basis there would appear to be no need to invoke a “multiple” nucleation event with a number of parallel laths forming a mutually accommodating system. ACKNOWLEDGMENT
The authors are grateful to Dr. P. R. Swarm for his comments and to Professor J. G. Ball for laboratory facilities. The work was started while one of us (G. C. W.) was at the Metallurgy Division, A.E.R.E., Harwell and he is grateful to U.K.A.E.A. for the award of a research fellowship and N.R.C. Canada for financial support. K. E. E. acknowledges support in the form of a Science Research Council award. REFERENCES 1. K. E. EASTERLINQand H. M. MIEEK-OJA, Acta Met. 15, 1133 (1967). 2. K. E. EASTERLIN~and P. R. SWANN, Proc. Cmf. Mechan&m of Phase Transformations in Crystalline Solids. Inst. Met& (1968). 3. J. W. CHRISTIAN, Proc. Cmf. Me&a&am of Phuae Transfomtions in Crystalline Sobid8. Inst. Metals (1968). 4. J. W. CHRISTIAN, Proc. Conf. Mechanism of Phase Transfomnations in Cyetalline Solida, discussion to Ref. 2. Inst. Metals (1968). 5. P. B. HIRSCH,J. Inst. Metals 86, 7 (1957).
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6. N. E. RYAN, Proc. Con!. Mechanism of Phase Traneformatione in Crystalline Solids, discussion to Ref. 2. Inst. Metals (1968). 7. V. K. LINDROOS and H. M. MIEKK-OJA, J. Inst. Metals 93, 513 (1964-66).
8. R. RHTY, V. LINDROOS, A. SAARINEN, J. FORSTBN and H. M. MIEKK-OJA, J. scient. In&rum. 45, 367 (1966). 9. H. 0. ALI-EL-AWAD, R. ADAMS and P. R. SWASN, to be published. 10. A. B. GRENINGER and A. R. TROIANO, Trans. Am. Inst. Min. metall. Engrs 185, 590 (1949). 11. J. W. CHRISTIAN, The Theory of Transformations in Metals and Alloys. Pergamon (1965).
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12. K. E. EASTERLINQ, Tekn. lis. Thesis, Finland Technical University ( 1965). 13. T. NIINIKOSKI and J. FORST~N, private
(1968).
communication
14. J. D. ESHELBY, Prog. Solid Mech. 2, 89, (1961). 15. J. W. CHRISTIAN,Acta Met. 6, 377 (1968). 16. J. W. CHRISTIAN, Acta Met. 7, 218 (1959). 17. L. KAUFMAN, Acta Met. 7, 216 (1959). 18. J. W. ESHELBY, Proc. R. Sot. A241, 376 (1957). 19. E. R;~s~NEN, Tekn. lis. Thesis, Finland Technical University (1967). 20. L. KAUFMAP; and M. COHEN, Prog. Metal Phys. 7, 165 (1958). 21. M. F. ASHBY, Phil. Mag. 14, 1157 (1966).