TWINNING AND MARTENSITIC TRANSFORMATION

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TWINNING AND MARTENSITIC TRANSFORMATION

J. Christian

To cite this version:

J. Christian. TWINNING AND MARTENSITIC TRANSFORMATION. Journal de Physique Collo-

ques, 1974, 35 (C7), pp.C7-65-C7-76. �10.1051/jphyscol:1974705�. �jpa-00215861�

TWINNING AND MARTENSITIC TRANSFORMATION

J. W. CHRISTIAN

Department of Metallurgy and Science of Materials, Oxford University, England

RBsumB. -

On peut regarder le maclage de deformation et la transformation de martensite, tous les deux, comme des f a ~ o n s de dbformation, et on passe en revue les modkles qui rapportent la formation de germes et la croissance de macles et de martensite aux dislocations et aux fautes d'empilement. Les investigations recentes, thkoriques et experimentales, montrent qu'il est possible que les macles en cristaux c. c. aient ses origines dans les dislocations vis cornme des fautes de trois ou quatre plans, et que la nucleation des boucles successives de dislocations de maclage ou de trans- formation, assistee par contraintes, soit beaucoup plus facile que la thBorie Blastique indiqute.

Quelquefois, la propagation de glissement a travers des interfaces coherents demande la combinai- son des dislocationsr~ticulaires par groupes, et il est possible que le plan et la direction de glissement nouvelle soient exceptionnels

;

les considerations similaires appliquent aux interactions macle avec macle. On discute la relation entre la structure interfaciale des lames de martensite semi- coherente et les monocristaux, les cristaux avec fautes et les cristaux maclbs, et on decrit les travaux recents au sujet de martensite assist& par les contraintes ou formee par les deformations.

Abstract. -

Both deformation twinning and martensitic transformation may be regarded as modes of deformation, and models which relate the nucleation and growth of twins and martensite plates to dislocations and stacking faults are reviewed. Recent experimental and theoretical work indicates that b. c. c. twins mays originate from screw dislocations as three- or four-layer faults, and that the stress-assisted nucleation of successive loops of twinning or transformation dislocation may be much easier than indicated by elastic theory. The propagation of slip across coherent interfaces sometimes involves the combination of groups of individual lattice dislocations and the new slip system may be non-conventional

;

similar considerations apply to twin-twin intersections.

The relation of the interface structure of semi-coherent martensite plates to single crystal, faulted single crystal and twinned products is discussed, and recent work on stress-assisted and strain- induced martensite is described.

1.

Introduction.

- A deformation twin is produced by a homogeneous simple shear of the lattice, and the region twinned undergoes a macroscopic change of shape which is described by the same simple shear. For a shear of magnitude s in the direction of the unit vector e on the plane K,, the successive displacements of parallel K , planes of spacing d are each given by

b,

⁼

sde

where b, is generally smaller than the interatomic distance. The displacement b, applied across a single K, plane defines a shear fault or translational twin [I]

of the structure, and the possibility that a monolayer twin of this type may exist as a metastable defect is readily envisaged, at least if K , is rational. The fault may also be regarded as a mistake in the stacking sequence of the K, planes and may thus be detected by appropriate diffraction techniques

;

the familiar intrin- sic (1 V) faults of the f. c. c. structure are monolayer twin faults of this type. It follows that any twin may be formally described as an ordered stack of such faults a t intervals of d, where d is either the smallest separation of the lattice K , planes or some multiple of this sepa-

ration. Moreover, since the monolayer fault has an edge which may be regarded as a glissile dislocation of Burgers vector b,, the formation of the twin may be accomplished by the passage of a linear defect (a twinning dislocation) with this Burgers vector through the successive K, planes. In a twin which is more than a few layers thick, the twinning dislocation is alter- natively described as a step in the coherent K , inter- face. However, it should be emphasized that the above geometrical description is purely formal and that monolayer faults and twinning dislocations have not been detected in most structures which twin.

The simplest type of martensitic transformation involves lattice changes which are closely similar to those in deformation twinning. Provided the two crystal structures possess a mutual lattice plane on which they fit together without distortion, the lattice change as the product phase is produced is again iden- tical with the macroscopic change of shape, and both changes may be regarded as a combination of a simple shear on the plane of fit with an expansion or contrac- tion normal to this plane. This type of martensite will be described as fully coherent ; the lattice deformation

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974705

C7-66 J. W. CHRISTIAN

is an invariant plane strain and the successive displa- cements of the invariant planes are given by

bT

=

gde

where g now specifies the magnitude of the strain and the direction e is no longer confined within the inva- riant plane. Once again a displacement b, applied across a single plane defines a generalised fault of the parent lattice, although this is no longer necessarily a pure shear fault

;

the (1 V) fault of the f. c. c. structure may be alternatively regarded as a monolayer of the h. c. p. structure, and this may be preferable to regard- ing it as a twin monolayer if the spacing of the close packed planes across the fault is measurably different from that in the unfaulted lattice. A step in the cohe- rent interface of the martensite plate now has a Burgers vector given by eq. (2) and is called a transformation dislocation [2, 31. Motion of a transformation disloca- tion causes thickening of the coherent martensite crystal.

The conditions for two lattices differing only in orientation to be related by a simple shear can be satisfied in an enumerable infinity of ways, although in practice only a limited number of types of twin are encountered in any given structure. The corresponding conditions for two non equivalent lattices to be related by an invariant plane strain are much more restric- tive

[4]

and since these conditions are expressed in terms of the relative lattice parameters of the two structures, they can be satisfied only coincidentally or in closely related structures such as f. c. c. and c. p. h.

Most martensites are thus not of the fully coherent type described above but have instead a semi-coherent habit plane interface which is nevertheless glissile when subjected to an imposed chemical or mechanical driving force. There is then a distinction between the lattice and the shape deformations, and only the shape defor- mation is an invariant plane strain. The concept of a monolayer of the plate as a fault on the irrational habit plane is not generally useful, although the possibility that the original nucleus is developed from stacking faults of the parent structure still exists. The formal model of the semi-coherent interface now requires it to contain primary dislocations which must have Burgers vectors which are corresponding lattice vectors of the two structures if both structures are to remain unfault- ed. Cases are known where the product structure consists of a closely spaced stack of faults, implying that the interface dislocations have Burgers vectors which are not lattice repeat vectors. A more common situation is that the product structure is twinned on a fine scale.

The three main types of lattice change which are to be discussed are shown schematically in figure 1, which emphasizes the distinction between the deformation of the lattice and the macroscopic change of shape.

Because of the change of shape, all three types of transformation will relieve the potential of an exter-

FIG. 1, - Schematic illustration of lattice relations and shape deformations : a) Deformation twinning. b) Coherent marten-

site. c) Semi-coherent martensite.

nally applied stress, and hence may be considered as modes of deformation.

2. Geometrical relations and structures of fully coherent interfaces.

^-

The K, interface of a defor- mation twin is rational in compound and type I twins and irrational in type I1 twins. Irrational interfaces may also be expected, at least in principle, if the definition of deformation twinning is widened [5], to include shear deformations in which the mutual orientations of the 'two lattices are not restricted to those rotation and reflection operations which are described in the classical theory of twinning. In simple cases, the twinning shear relates all the lattice points of the two crystals, and hence all the atoms if the primitive unit cell contains only one atom. When there is more than one atom per lattice point, as for example in h. c. p. structures, each atom in the motif unit may be placed on a separate sub- lattice and these interpenetrating lattices combine to give the overall or multiple lattice structure. The atom movements in twinning may then be analysed formally into a simple shear, common to each interpenetrating lattice, plus translations of these lattices with respect to each other. The translations do not contribute to the macroscopic shape change and are called shufJs.

Shuffles may also arise if the twinning shear relates only some integral fraction of the lattice points of the two crystals

;

this happens in type I twinning if the smallest lattice vector in the q, direction crosses more than two lattice K, planes, and in type I1 twinning if the smallest vector parallel to q, crosses more than two K, planes. Lattice shuffles of this type could, in principle, give rise to additional twinning modes even in single lattice structure but in practice they seem to occur only when structure shuffles are required. The rules govern- ing the observed deformation twinning modes are apparently avoidance of shuffles if possible and minimi- sation of the shear magnitude s.

With respect to an origin in the K, interface and a

co-ordinate system defined by the vector triad ai and

reciprocal basis a', the lattice points in the parent

structure may be specified by the vectors u

⁼

ui ai and

the lattice points (or some integral fraction of these

points) in the twin are then given by the vectors v

= vi

a, where

The tensor S = S; represents the simple shear and t is an additional translation of all the twin lattice points with respect to the parent lattice. The vector t can be treated as effectively constant except for sites close to the interface which undergo additional inhomogeneous relaxations. The possibility that t is non-zero has frequently been ignored, and fully coherent interfaces are usually represented as in figure 2 with a plane of

FIG. 2. - Idealised models of coherent interfaces assuming lattice coincidence [7] : a) b. c. c, twin, b) h, c. p. twin, c) f. c. c.- h. c. p. interface. Open and full symbols represent atoms on two successive planes parallel to the plane of the figure. In the h. c. p.

structure, circles and squares represent atoms on non-equi- valent sites.

lattice points (atoms in the simpler cases) common to the two structures. Recent work, however, shows that .this may be unjustified

;

the value o f t becomes impor- tant when the structure of the interface is being considered, and it may also be detected by diffraction contrast effects in the electron microscope [6].

In terms of the unit normal, n

=

ni a', to the K, plane and of other quantities already defined

S!

_J =

6:

_J

+ se' nj . _(4>

There is also a conjugate or reciprocal twinning mode with the same shear magnitude but in which the roles

of the K, and K, planes and the q, and

q,

directions are interchanged. There are simple analytical relations between

se

and n and the unit vectors

Z

and n parallel to q2 and normal to K, respectively [7]. The tensor S gives the shape change when a deformation twin is formed, but eq. (3) also has a more formal interpreta- tion since it may be used to specify the relative atomic positions for all twins, including annealing twins. Of course, S has no physical significance for annealing twins, and may be replaced by any of the appropriate rotations R carrying the parent lattice into the product, but whatever representation is used, t cannot necessa- rily be set equal to zero.

The simplest twin fault has a displacement vector given by b, of eq. (I), but the possibility of mono- layer faults with more general fault vectors b, + t must also be considered. Vitek [8] made calculations of the relative energies of infinite shear faults of this general type for the { 11 1 ) planes of the f. c. c. structure and the { 110 ) and { 112 ) planes of the b. c. c. structure using various two-body interatomic potentials and allowing full relaxation of the atomic positions around the fault. His results were obtained in the form of three-dimensional plots of energy y as a function of the two components of the displacement in the plane of the fault. As expected, this so-called y-surface for the f. c. c. { 11 1 ) planes has a local minimum at approxi- mately the position corresponding to a fault vector of type

-

a < 112 >, which is the Burgers vector of a

6

Shockley partial dislocation (see Fig.

3).

This means

FIG. 3. - A computed y-surface for the (1 11) plane of a f. c. c.

lattice [S].

that large 1 V faults are possible metastable defects, in agreement with experiment, and presumably also implies that deformation twins might grow from such monolayer faults. A more tentative conclusion is that the displacement vector t for a { 11 1 ) twin boundary is zero .

The ~esults for { 110 ) and { 112 ) faults in the b. c. c.

structure were rather unexpected. No local minima,

other than those corresponding to the perfect lattice

were found in either y-surface. The results appropriate

to twin faults are those for { 112 ) planes (Fig. 4) and

they show that a wide monolayer twin fault would be

mechanically unstable and thus could have only a

IRISTIAN

FIG. 4.

-

A computed y-surface for the (112) plane of a b. c. c.

lattice [8].

transitory existence. Vitek has emphasized that although the actual energies found in these calculations are sensitive to the form of the potential, the shape of the y-surface, and in particular the absence of local minima, is probably a general characteristic of the crystal structure.

If a monolayer twin is unstable, how many layers does a metastable twin of minimum thickness possess

?

By considering a n-layer fault and allowing full atomic relaxation in the various layers (Fig. 5) Vitek

[9]

found

FIG. 5. - A multi-layer fault [9]. The displacements of the planes 1

-

n are variable, giving 3 n degrees of freedom.

a metastable configuration with only three layers given displacements of

_f4_

₁₂ < T i 1 >,_a < - i l l > a n d 2 < ^{i i l} >

6 12

with respect to the preceding layers. The same calcula- tion applied to an infinite twin produced a similar result, namely the displacement across the first layer is approximately

- a

< ¹¹¹ > and across all succeeding

12

layers is

- a

< ¹ >. Since this latter displacement is 6

the expected Burgers vector of the twinning dislocation, given by eq. (I), the calculation shows that for the

b. c. c. twin, the vector

t

of eq.

(3)

is

- a

< 117 >

^;

this 12

result has recently been confirmed in a separate calculation [lo]. The twin interface (Fig.

6a)

may thus be regarded as a combination of the fully coherent (coincidence site) model (Fig. 2a) with a shear fault.

However, as illustrated in figure 6b, the structure retains a high degree of symmetry. In particular, the atomic configuration at the interface is the same when viewed from either the matrix or the twin.

FIG. 6.

-

Computed b.c.c. coherent (112) twin interface.

a) To illustrate displacement from figure 2a 191. b) Same struc- ture re-drawn to emphasize symmetry [lo].

For any lattice, there is a plane of lattice sites common to the two crystals when t

=

0 and this is also, of course, a plane of the coincident site lattice. How- ever, if there is more than one atom per lattice site, geometrical fit of all the atom sites may not be possible at the interface, so that even with

t =

0 there must be some distortion of the structure near the interface. For a double lattice structure, the simplest assumption is that lattice sites at centres of symmetry fit exactly along the interface

;

in effect, the K, planes each side of the interface may be regarded as oppositely puckered atom planes, and the interface is a planar net. Figure 2b shows this idealised structure for a h. c. p. ( 1012 ) twin and it should be noted that the atoms at the inter- face have completed one-half of the shuffle displace- ment. Preliminary calculations have indicated interface structures of this type for twins in beryllium [ll].

The formal theory of fully coherent martensites is similar to that of deformation twinning. Eq. (3) still applies with

S;

=

6f +

gei

nj . ( 5 )

Provided one invariant plane with normal n can be

found for a given pair of lattices, a second plane n must

also exist, so that there is a conjugate transformation

mode. The geometrical relations between the two

modes are slightly more complex than in the case of

twinning

[7,

121. When the invariant plane is rational,

simple coincidence lattice interfaces can be envisaged,

as shown in figure

2c

for the f. c. c.-h. c. p. transfor-

mation. There are, at present, no reliable calculations

on the actual atomic configurations at such interfaces.

3. Nucleation and growth of deformation twins and coherent martensites. - Spontaneous homogeneous nucleation of twins in defect-free regions under the influence of applied stress, or of martensite under the combined influence of chemical free energy and applied stress, is very improbable unless s or g is unusually small [13]. Most theories of nucleation are thus concerned with defect configurations and more specifi- cally with the dissociation of single dislocations or the mutual interactions of a group of dislocations. The structural connection between a 1 V fault of the f. c. c.

structure and a monolayer of h. c. p. was first used in a description of the f. c. c.

-+

h. c. p. transformation in cobalt 1141 and was also applied to f. c. c. twinning.

Many later models have been evolved for f. c. c. and b. c. c. twinning, and usually depend on the dissocia- tion either of a glissile dislocation in its plane, or of a primatic dislocation into glissile elements. These models, in contrast to the thermodynamic treatments of homogeneous nucleation, do not distinguish clearly between the nucleation and growth stages. The critical nucleus configuration is not identified, or is considered to be a monolayer fault of unspecified diameter, and attention is concentrated on the way in which the fault first forms and subsequently thickens.

A combined nucleation and growth model for b. c. c.

twinning due to Cottrell and Bilby [15] introduced the concept of the pole mechanism for growth and had a very considerable influence on the development of the theory. Growth from a monolayer fault, as envisaged in this model, is geometrically possibly only for the b. c. c.

lattice and the initial dislocation dissociation is moreover energetically unfavourable. Sleeswyk [16]

suggested that the nucleus is a three-layer fault formed by the dissociation of a screw dislocation on three successive (112) planes according to the schematic reaction

:.

a -

[111]

⁼ !f

[111] + a [lll] + a [I1111 .

2 6 6 6

In view of the atomistic calculations of the y-surfaces for b. c. c. structures, this proposal seems more acceptable than growth from a monolayer fault, although the structure of figure 6 implies that the first nucleus might perhaps be a four layer fault formed by the dissociation

Some support for this model is available from both theory and experiment. Calculations of the core struc- ture of a screw dislocation and its behaviour under applied stresses [17, 181 have given configurations very similar to those of a twin nucleus, although in almost all cases so far investigated the dislocation has even- tually moved before a wide three- or four-layer fault

has formed. It is worth noting that the core structures and their changes under stress appear to be rather sensitive to the appropriate y-surface and to the F-surface where the restoring force F

⁼

- grad y. The dimensionless quantities y*

= ylpa

and F *

= F / p

where

p

is the shear modulus resolved in the direction of the fault displacement and a is the lattice parameter have been used to compare hypothetical b. c. c. metals with different strengths of binding [17-191 and these factors may account for the very variable twinning tendencies of real b. c. c. materials.

Mahajan [20] has made an electron microscopic investigation of the early stages of development of deformation twins in a molybdenum-35>% rhenium alloy which twins very readily. He found that fine twins and faults bounded by screw partials are inter- spersed with glide dislocations of screw character.

Figure 7 shows a well developed twin preceded by a few tiny twins and some screw dislocations. The taper- ing twin gives rise to fault-type fringes (Fig. 7a) and under different diffraction conditions (Fig. 7b) it is seen

FIG. 7. -Electron micrographs of twins and dislocations in a Mo-35 % Re alloy [19], a) g = 101, micrograph plane

-

^(111).

b) g = 312, micrograph plane

-

^(111).

that the steps correspond to twinning dislocations.

Comparison of the contrast with four diffierent g vectors showed that the Burgers vector of these twinning dislocations is indeed parallel to that of the lattice screw dislocations. Mahajan suggested that three-layer faults originate from screw dislocations in a manner similar to that envisaged by Sleeswyk, and that macroscopic twins might result when three layer twins grow into each other. The mechanism implies that some slip probably precedes twinning, and there is experimental evidence [21, 221 in support of this assumption.

Orowan's picture [23] of a tapering or lenticular twin incorporates closed loops of twinning dislocation on successive K, planes, and the usual model of edgewise growth simply involves the expansion of these loops. A slightly more complex model for the growth of b. c. c.

twins was developed by Sleeswyk [24] in terms of

emissary dislocations as shown in figure 8. Although

there is some convincing evidence for emissary slip, it is

C7-70 J. W. CHRISTIAN

MATRIX

MATRIX

FIG. 8. - Sleeswyk's model of emissary dislocations [24].

quite probably that it is mainly a special form of accommodation slip, produced when a twin is stopped within a crystal, rather than a normal mode of growth.

The thickening of a twin with a low energy coherent K, interface presents problems similar to those encountered in the theory of crystal growth on low- index planes, and thus it seems inherently probable that the growth mechanism will utilise the topological properties of the dislocation node produced when a lattice or pole dislocation crosses the interface. The pole mechanism for either a twin or a coherent marten- site plate may be developed by supposing a lattice dislocation of Burgers vector b, to be present in a region which is subsequently partly transformed. In the transformed region, the dislocation now has a Burgers vector b,

=

SbA where, from eq. (5)

where b , is the Burgers vector of the appropriate twinning or transformation dislocation. If this disloca- tion now rotates about the fixed lattice dislocations, the node is displaced a distance d

=

bh ni for each complete revolution, resulting in growth of the twin or martensite plate. Rotation in the opposite sense displaces the node through b; ni and the twin shrinks.

Provided b, and b, are both lattice dislocations, the above theory is sufficient to ensure the operation of the generating node and the topological property is independent of whether the configuration is obtained by assuming the interface to move past the dislocation or the dislocation to glide across the interface (see below).

If d represents the smallest spacing of the lattice planes parallel to the coherent interface, i. e. if the Burgers vector of the pole dislocation connects lattice points in adjacent planes, the mechanism will effect the transformation of all the lattice points, but in other cases some lattice shuffles may be required. Conversely, if the twinning or transformation mode involves lattice shuffles, it will not generally be possible to choose both b, and b, as lattice vectors unless d is some multiple of the smallest spacing. When this is the case, the dislocation defined by eq. (1) or (2) are called zonal twinning or transformation dislocations and are

distinguished from elementary twinning or transfor- mation dislocations of minimum step height. The zonal dislocation has a step height which represents the minimum distance at which the atomic configuration of the interface is repeated

;

in twinning with rational K,, for example, the structure repeats at intervals of q lattice planes (q odd) or

-

1 q lattice planes (q even).

2

where q is the number of K, planes traversed by a primitive lattice vector parallel to qz. A zonal disloca- tion may, of course, dissociate into elementary twin- ning or transformation dislocations, but these will be separated by interface regions of higher energy than the minimum energy

;

the dissociation is analogous to- that of a lattice dislocation into partials separated by stacking faults. Figure 9 shows schematically [12] a n elementary and a zonal twinning dislocation for q

⁼

3.

FIG. 9.

-

Elementary and zonal twinning dislocations for q = 3 [12]. Equivalent cells ABCD, ABC' D' are distorted by formation of an elementary step, but not by a zonal step.

P + P' and Q ⁺ Q' indicates one possible shuffle mechanism.

A suitable generating node for f. c. c. twinning defined as in eq. (9) is

where all Burgers vectors are referred to the conven- tional cubic axes of the parent crystal. A similar node for the f. c. c.-h. c. p. transformations when the h. c. p.

plane has axial ratio a is [2]

The vector on the left of this equation is not a lattice vector of the cubic structure but is

c

[OO. 11 of the h. c. p.

structure. The pole dislocation in the cubic structure could have been written

- 1

a [I 121 but a combination of

2

two lattice vectors of type

-

1 a < 101 > is more pro- 2

bable, giving a four-dislocation node. The last vector of eq. (1 1) represents the transformation dislocation and reduces to

-

a [I 121 when a has the ideal value of (813)'''

6

As described above, the origin of the pole disloca- tions is separate from the nucleus of the twin or mar- tensite, and there is then no difficulty in finding suitable pole dislocations. However, in most models, the pole dislocation and the fault nucleus originate from the same lattice dislocation and the geometrical relations are then not so simple. Sleeswyk [25] has recently criticised the Cottrell-Bilby model for b c. c. twin- ning [I51 and the Venables model for f. c. c. twin- ning [26], on the grounds that the pole dislocations in the twins are imperfect.

Alternative mechanisms of thickening involve either the repeated dissociation of separate lattice dislocations on successive K, planes or the successive nucleation of loops of twinning or transformation dislocation, either at the extremities of the growing plate (e. g. at free surfaces, grain boundaries, etc.) or internally. Repeated dissociation is geometrically possible, although impro- bable, in b. c. c. crystals when lattice screw dislocations glide up to an existing fault or twin which intersects the glide plane but is parallel to the Burgers vector. A rather similar proposal has been made for the nuclea- tion and growth of a f. c. c. twin [27]. In this case, an extended dislocation on a { 11 1 ) slip plane encoun- ters a large fault or twin on the other ( I l l ) plane which contains its Burgers vector, and redissociates in the plane adjacent to the interface to form a n extrinsic fault. Separation of the partials bounding this fault then corresponds to thickening of the twin, and the process can be repeated.

Thickening by homogeneous nucleation of suc- cessive closed loops of twinning dislocation may be treated by elastic theory and is analogous to the homogeneous nucleation of lattice dislocations or the two-dimensional nucleation of layers on close-packed crystal surfaces. Unless the Burgers rector of the twinning dislocation is very small, as, for example, in indium-thallium [28] or gold-cadmium [29] alloys, the probability of nucleation at the observed twinning stresses or chemical driving forces is negligible.

However, this conclusion rests on the assumption that the interface region possesses the elastic properties of the matrix, and several authors have pointed out that this may be erroneous [30].

Preliminary calculations by Yamaguchi and Vitek have recently indicated that an isolated twinning dislo- cation on a b. c. c. crystal may spread over many

atomic distances to form a diffuse step and that spontaneous formation of such steps may occur at stresses much below those expected from elastic theory. These conclusions must b e regarded as extre- mely tentative at present, but there are nevertheless some experimental observations which point in the same direction. Thus figure 10 shows electron micro- graphs obtained by Pumphrey and Bowkett [31] in which

-

I a [I011 lattice dislocations dissociated into

2

Shockley partials are visible in or near a (111) twin boundary in an austensitic stainless steel. According t o conventional elastic theory, the separation of the par- tials would correspond to a fault energy of - ^{1 mJrn-'}

which is only - 1160th of the reported matrix fault energy

;

the authors therefore suggest that the matrix has unusual properties close to, the coherent twin interface.

There have been few observations of transformation dislocations in coherent martensitic interfaces such as cobalt, but measurements of the macroscopic shape change in cobalt and its alloys have been interpreted as indicating that the transformation dislocation has the same Burgers vector in a relatively large number of successive planes. This would be in accord with the pole, mechanism, but not with homogeneous nucleation.

4. Propagation of slip and twinning across coherent interfaces.

-

Incoherent interfaces are effective bar- riers to slip but lattice dislocations may cross coherent twin or martensite interfaces. If the interface contains the direction of a slip vector, dislocations with that Burgers vector must meet the interface in screw orien- tation and can cross slip into the other crystal on any convenient slip plane without leaving a defect in the boundary. When this condition is not satisfied, the slip in one crystal can still be continued in the second crystal on any plane which meets the first plane edge t o edge in the interface, but conservation of the Burgers vectors then requires that a linear defect is also left in the interface. The simplest case is that considered above when the two Burgers vectors of the slip disloca- tions, b, and b,, are corresponding vectors of the lattice deformation, i. e. are related by S, and the slip planes in the two crystals are then corresponding planes which automatically meet edge to edge. If a finite length of lattice dislocation glides across an interface in this way, the two limiting points are generating nodes of the type discussed above and are joined by a length of twinning or transformation dislocation. The whole configuration forms a double ended source which is topologically suitable for subse- quent growth of one crystal at the expense of the other, provided the lattice dislocations are subsequently pinned. Whether double ended pole sources are actually formed in this way is uncertain.

The dislocation can only glide into the second

crystal if the new Burgers vector is a lattice vector of that crystal, and the general conditions for this when the Burgers vectors are connected by S were enu- merated by Sax1 [32]. In type I twinning, for example, the step height of the defect left in the interface must be an integral multiple of qd (q odd) or of

-

1 qd (q even)

2

where d is the minimum spacing of the K, planes. This means that the remanent defect must be a zonal twinning dislocation rather than an elementary twinning dislocation, and the condition is equivalent to that stated above for formation of a practicable generating node. There are equivalent considerations for rational coherent martensite boundaries. Elemen- tary twinning or transformation dislocations cannot be defined when n is irrational, but in type I1 twinning, conditions forb, to be a lattice vector may be expressed in terms of the number, q, of rational K, planes traversed by a primitive lattice vector in the 11, direction.

When a single lattice dislocation is unable to cross the interface, it is always geometrically possible for a group of like dislocations to cross simultaneously, forming a single dislocation with a latticz Burgers vector in the other crystal. An example of this is slip across { 10i2 ) twins (q

=

4) in h. c. p. crystals. Only one of the three < 1120 > slip vectors of the matrix is transformed into a lattice vector of the twin, and there is experimental evidence that a moving twin boundary pushes single dislocations of the other types ahead of itself, rather than incorporate them as fauits [33]. Pairs of dislocations, however, may glide from matrix basal or prism planes to give twin dislocations with Burgers vectors of type

^-

1 < ^aaZa3c > moving on { 1070 ) or

3

{ 1122 ) planes respectively [34]. Despite the unusual nature of the slip systems in the twin, this process has actually been observed by Tomsett and Bevis [35] for basal slip in thin foils of zinc (see Fig. 11).

Dislocations meeting the interface, even in non-screw orientation, are not geometrically constrained to continue on the plane produced from the primary slip plane by S

;

any plane which contains the line of the dislocation is a possible new slip plane, and a new slip vector is obtained by adding a lattice vector of the second crystal to both SbA and b,, i. e. to each side of eq. (9). However, there are unlikely to be many slip systems of this type which have low indices. Clearly, geometrical continuity of the slip systems is not suffi- cient to ensure that dislocations will actually glide across the coherent boundary. The new slip system may not be a usual system, as illustrated by figure 1, or it may carry a rather low resolved shear stress.

The propagation of a deformation twin (A) across an existing coherent twin (B), forming a secondary twin (C), may be treated by examining the compatibility of the various shears, or equivalently by extending the above theory to the case when the crossing dislocation

FIG. 10.

-

Dissociated dislocations in or near twin boundary in stainless steel [31]. In (a) only grain 1 is diffracting whereas in (b) both grains are diffracting and g = 202 for grain 1. The Burgers vectors of the partials, a and b, are

2

^[I 121 and [21l] referred to grain 1.

6 6

FIG. 11.

plane of

- Electron micrograph showing slip trace on (0001) zinc, crossing (1702) twin bouhdary, and continuing

on (1100) plane of twin [35].

is a twinning partial of both lattices. General conditions

were first given by Cahn [36] who concluded that the

traces of A and C in B must be parallel and the direc-

tion

11,

and magnitude and sense of shear in A and C

must be the same. He considered specifically the case in

which the shear direction of A and C is parallel to the

line of the interaction so that B undergoes no displace-

ment normal to its K, plane. As shown in figure 12,

the K, planes of A and C are then mirror images in

the K, interface of B. In terms of the dislocation

FIG. 12.

-

Types of twin-twin interaction [37].

description, the twinning dislocations of A meet the matrix-B interface in screw orientation and hence can continue to glide without leaving steps at the interface,

Liu [37] pointed out another geometry which also satisfies Cahn's conditions. The g, directions of A and C are no longer contained in the matrix-B interface, but it is now necessary that the K, planes of A and C be parallel. This situation is also shown in figure 12

;

the crossing twin A is undeviated but B is displaced.

Individual twinning dislocations now each produce a step in the interface as they cross it, and the accumula- tion of these steps on successive planes produces the rotation of the interface.

Many authors have examined theoretically the conditions under which slip or twinning might pro- pagate across coherent twin boundaries in particular structures and recent experimental work has been reported on b. c. c. [38, 391, f. c. c. [40] and h. c. p.

[35, 411 metals. It has also been suggested that an unusual { 5, 8, 11 ) twinning mode detected in some

FIG. 13. - Optical micrograph of a twin-twin interaction in a Co-8 % Fe alloy [40].

b. c. c. and b. c. t. martensites arises because such twins may propagate undeviated across the closely- spaced { 112 ) transformation twins of the martensite, at least in the cubic case. Another possibility is that the deformation in one twin crossed by another may be produced by slip, so that Cahn's conditions do not apply. Figure 13 shows an optical micrograph of the surface of a f. c. c. cobalt-iron alloy in which a twin on ( i l l ) has crossed a twin on ( i l l ) without apparently forming a secondary twin [40]. The slip traces are not visible in the ( i l l ) twin, but Mahajan and Chin consider various ways in which this slip could occur on the ( i l l ) plane of the twin which is parallel to the (1 13) matrix plane. The micrograph also illustrates the continuity across the twin interfaces of slip on the (1 11) and (1 11) matrix planes.

Sleeswyk [24] used his theory of emissary disloca- tions to demonstrate the equivalence of slip and twin propagation across b. c. c. { 112 ) twin interfaces, and emissary arrays have been investigated by several

FIG. 14. - Dislocation arrays associated with twins in Mo-35 % Re alloy [43]. a) Simple emissary array, g = 110, plane of micro- graph

- ^(715).

b) g =-110, plane of micrograph ~ ( 1 1 3 ) . c) Complex array, g = 101, plane of micrograph

-

^(111).

6

C7-74 J. W. CHRISTIAN

authors [38, 42, 431. Figure 14 shows electron micro-

graphs of three arrays of varying complexity considered by Mahajan [43]. In the simplest array, the Burgers vector of the dislocations +is' parallel to the displace- ment vector of the twin and-slip is confined to the region ahead of the twin, whereas in tbe thi'?d case the slip pattern is very complex. Mahajan concludes that the emissary sub-structures ,evolve to relax the stresseq around the tips of twins which have ceased to grow, and that they are probably formed first by a coalescence reaction of three closely spaced twinning disloca- tions [44] i. e. by the reverse of eq. (6).

5. Semi-coherent martensite.

-

The formal theories of martensite crystallography [3, 45-61 will not be reviewed

;

they involve a distinction between the lattice deformation S and the shape deformation E which may be expressed either as

where G is the so-called lattice invariant deformation and S,, S, are different lattice deformations leading to twin-related products with volume fractions (1 - f ) and .$ If the martensite plate is a single crystal, the dislocation content of the interface may be defined by the Bilby equation r471

where b, defined in the parent lattice, is the net Burgers vector of the interface dislocations crossing any inter- face vector x. This equation is essentially identical with the basic equation of Bollmann's 0-lattice theory [48]

;

the relation between this theory and the formal theory of martensite is an interesting topic but cannot be pursued here.

When, as is usually assumed, the lattice invariant deformation G is a simple shear, the dislocation content of the interface by (14) reduces to a single set of parallel lines which glide on corresponding planes of the two lattices as the interface is displaced conservatively in one or the other direction. If both lattices are defect free, these dislocations must have Burgers vectors which are corresponding lattice vec- tors of the two crystals, and their mean spacing is then fixed by the magnitude of the shear represented by G . This is illustrated in figure 15a. However it is also possible to obtain the same net Burgers vector from an array of interface dislocations, the individual Burgers vectors of which are not lattice vectors. One of the crystals must then contain an evenly spaced (on average) array of stacking faults, as shown in figure 15b.

Martensites of this type will obviously not be found in practice unless the fault energy is very low

;

an example

FIG. 15. - Schematic models of semi-coherent martensite [12].

a) Single crystal martensite. b) Faulted martensite. c) Twinned martensite.

FIG. 16. - Electron micrograph of faulted martensite in Cu- 12

%

Al alloy [49].

It is well known that twinned martensites (eq. (13)) may be also represented macroscopically by an appropriately chosen lattice invariant deformation.

Figure 15c illustrates,this in model terms

;

the faults of figure 156 are clustered together on adjacent planes rather than evenly spread, and they then coalesce into a set of twins. The two cases can, of course, be readily distinguished in electron micrographs

;

figure 17 shows

- -

occurs in co~~er-aluminium alloys 1 4 9 ~ and is shown

_{FIG. 17.}

-

Electron micrograph of twinned marfensite in Fe-

in figure 16.

21.7

%

Ni-1

%

C alloy [501.

a twinned martensite plate in an iron-nickel-carbon nucleation of a-plates

;

figure 18 shows a well-known alloy [50]. Semi-coherent martensites are very often example due to Venables [56]. A double-shear model twinned on a fine scale. for the f. c. c.

+

b. c. c. transformation, due to Bogers Martensitic transformation is strongly influenced by and Burgers [571, seems to fit these observations, and externally applied shear stresses and it has also recently has been further developed by Olson and Cohen [58-91.

become evident that the internal stresses around pre- The Olson model is shown in figure 19

;

it involves two viously formed plates make a major contribution to

the subsequent course of the transformation

;

this is described as auto-catalytic nucleation [51]. Although transformation to another phase is not normally considered primarily as a mode of deformation, it is clear from work on TRIP steels and related alloys [52]

that the transformation strain may be of comparable importance to the intrinsic properties of the product structure in evaluating the utilisation of the martensitic transformation. It is thus relevant here to consider briefly the problem of nucleating a semi-coherent martensite plate.

The theory of homogeneous nucleation by thermal fluctuations is generally accepted as inapplicable to martensite formation. Various structural descriptions have been given for particular martensite transforma- tion in terms of dislocation dissociations, stacking fault intersections, etc. but almost all of these are approxi- mate insofar as the structure produced has incorrect lattice parameters and/or orientation, and the problem of matching at the interface is tacitly avoided. In the presence of an applied stress, the situation is even more complex and it is sometimes necessary to distinguish between stress-induced and strain-induced martensite.

Bolling and Richman [53] found that for some ferrous alloys there is a temperature range above M, in which yielding is initiated by the onset of martensite formation and in which the yield stress rises rapidly from zero at M, to a maximum value at the upper limit of the temperature range which they called M,". This type of martensite is referred to as stress-assisted and the observed yield stress-temperature relations are, consistent with the expected influence of elastic stresses on Ms. At temperatures above M," yielding begins with slip in the austenite and martensite forms only after some strain, so that the yield stress-temperature relation has its normal negative sign. This is called strain-induced martensite, and since it occurs at stresses much below those obtained by extrapolating to above Mg the yield stress in the stress-assisted region, it is concluded that strain centres appreciably affect the nucleation process. Eventually, at sufficiently high temperatures, martensite is no longer formed

;

the upper limit is designated M,. In some later papers 154-51 this usage is not followed and M, corresponds to the limit of stress-induced martensite formation.

In ferrous alloys of low stacking fault energy, the (presumably) coherent h. c. p. emartensite is formed prior to or in conjunction with a-martensite. I t has long been recognised that the intersections of two

&-plates, or of one &-plate with a f. c. c. twin or a slip band sometimes provide favourable sites for the

FIG. 18. - Formation of a-martensite at intersection of two e-martensite plates in stainless steel [56].

FIG. 19. -The Olson-Cohen model for nucleation of a-mar- tensite

[58].

intersecting sets of 1 V faults, spaced at intervals of

every two (111) planes and every three ( i l l ) planes

respectively, to produce a prismatic b. c. c. region with

a close-packed < 11 1 > ,, // [110],,, direction as the

long axis. The shear on (1 11) corresponds, in fact, t o

the production of an &-plate and the other shear might

be obtained by slip or from a highly faulted &-plateor

twin. Despite the intrinsic attraction of such models of

strain-induced nucleation, however, various difficulties

C7-76 J. W. CHRISTIAN

arise from the misfit of the structures and consequent accommodation stresses

[60] ;

for example there is a 2 % misfit along the long axis of the prismatic region of figure 19 in a typical case.

A semi-coherent martensite interface probably offers an appreciably higher resistance to the passage of dislocations than does a coherent boundary, but in principle continuity of slip should be possible if the defects are not too closely spaced. The transformation twins inside a martensite plate also make some contri- bution to the strengthening of the structure, but this is probably not a major effect. Twinned non-ferrous

martensites are relatively soft, and deformation twinning and slip have been observed in internally twinned martensite plates [61-21. The considerable strength of ferrous martensites is associated with the high supersaturation of interstitial solutes, but several individual mechanisms contribute to the overall strength, and these processes interact in a complex manner [63].

Acknowledgment.

-

The author wishes to thank Dr. V. Vitek for helpful discussions and comments on the manuscript.

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