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Crystallographic aspects of deformation twinning and consequences for plastic deformation processes in γ-TiAl
Acta mater. 48 (2000) 851±862 www.elsevier.com/locate/actamat
CRYSTALLOGRAPHIC ASPECTS OF DEFORMATION TWINNING AND CONSEQUENCES FOR PLASTIC DEFORMATION PROCESSES IN g-TiAi B. SKROTZKI Ruhr-University Bochum, Department of Mechanical Engineering, Institute for Materials, D-44780 Bochum, Germany (Received 6 August 1999; accepted 12 October 1999) AbstractÐMechanical twinning is an important deformation mechanism in the g-phase of near-g-TiAl alloys at room temperature as well as at high temperature. The present work examines the crystallography of twinning in the ordered L10 structure of the g-phase and quanti®es the resulting deformation for dierent crystallographic directions. In addition, the open question is addressed whether twin interfaces represent obstacles for glide dislocations. It is shown that one part of the possible glide dislocations remains completely unaected by the twin interface because their Burgers vectors lie in the twin plane. The other part of the dislocations has to change its glide directions while passing the interface, which can occur by suitable dissociation processes. Prerequisite is a suciently high applied stress. The reoriented twin may be more favorably oriented for some glide directions than its parent crystal and thus promote further dislocation plasticity. The impact of this eect on ductility depends on the volume fraction of twins. # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. ZusammenfassungÐMechanische Zwillingsbildung ist ein wichtiger Verformungsmechanismus der g-Phase von near-g-TiAl-Legierungen sowohl bei Raumtemperatur als auch bei hohen Temperaturen. Die vorliegende Arbeit beleuchtet die Kristallographie der Zwillingsbildung in der geordneten L10 Struktur der g-Phase und quanti®ziert die resultierende Verformung fuÈr verschiedene kristallographische Richtungen. DaruÈber hinaus wird die bisher oene Frage angesprochen, ob Zwillingsgrenzen Hindernisse fuÈr gleitende Versetzungen darstellen. Es wird gezeigt, daû der eine Teil der moÈglichen Gleitversetzungen durch die Zwillingsgrenze vollstaÈndig unbeein¯uût bleibt, da ihre Burgers-Vektoren in der Zwillingsebene liegen. Der andere Teil der Versetzungen muû jedoch beim Passieren der Zwillingsgrenze seine Gleitrichtung aÈndern, was durch geeignete Aufspaltungsprozesse geschehen kann. Voraussetzung dafuÈr ist, daû eine ausreichend hohe aÈuûere Spannung vorhanden ist. Der neu gebildete Zwillingskristall kann fuÈr einige Gleitrichtungen guÈnstiger orientiert sein als das Ausgangskorn. Der Ein¯uû dieses Eekts auf die DuktilitaÈt haÈngt vom Volumenanteil der Zwillinge ab. # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Intermetallic compound; Twinning; Crystallography; Interface; Dislocations
1. INTRODUCTION
Intermetallic near-g-TiAl alloys are regarded as new candidate materials for high temperature applications due to their low density combined with high strength and high Young's modulus which are retained at high temperature, and good corrosion resistance. TiAl technology has reached an advanced level and high temperature structural applications such as automotive engine valves, turbocharger rotors and turbine engine components are envisaged [1±3]. Technological aspects including structural evolution during thermomechanical processing, the eect of microstructure on mechanical properties and the in¯uence of alloying elements have been studied in great detail [4, 5]. In addition,
the creep behavior has been investigated intensively during recent years because possible applications at high temperature are often subjected to conditions where creep processes are inevitable [6±11]. Generally, the deformation behavior is of great interest due to the fact that TiAl alloys (as most intermetallic alloys) do not show much ductility at room temperature. Several studies have shown that beside dislocation glide and climb plasticity, mechanical twinning is an important deformation mechanism at room temperature as well as at high temperature for many near-g-TiAl alloys [7, 8, 12± 18]. The ordered L10 structure of g-TiAl is a CuAuItype superlattice with a tetragonally distorted f.c.c.
1359-6454/00/$20.00 # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 3 8 5 - 7
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SKROTZKI: DEFORMATION TWINNING
Fig. 1. (a) Tetragonally distorted f.c.c. unit cell (L10 superlattice) of g-TiAl. (b) True twin in a g-crystal with twinning plane K1 111 and shear direction Z1 112 : Filled symbols represent Ti atoms; open symbols are Al atoms situated above and below the drawing plane.
unit cell c=a 1:02 where the Ti and Al planes alternate in the c-direction [Fig. 1(a)]. The lattice shows large similarities to the f.c.c. structure. Thus, slip occurs exclusively on {111} planes either by dislocations or by twinning. Dislocation slip is governed by ordinary (perfect) dislocations with Burgers vector b0 1=2h110{ and by superdislocations with bs1 h101 and bs2 1=2h112 [14, 20]. Ordinary dislocations do not aect the order whereas slip along h011] moves atoms to antisites and two consecutive dislocations are required to maintain the order. Deformation twins are observed but only the four twin directions of h112 -type preserve the order structure [21]. These twins are generated by Shockley partial dislocations of type bp 1=6h112 and they produce no wrong ®rst neighbors. An intrinsic stacking fault (ISF) is created in the L10 structure and there exists only one of this type per octahedral plane. These so-called order twins are the only twin modes which were experimentally found [14, 22]. The other two Shockley partial dislocations of a (111) plane produce complex stacking faults (CSF) and the twin contains
{ Mixed notations hhkl ] are used to account for the tetragonality of the L10 lattice.
wrong ®rst neighbors on each plane. The energy to form a CSF or an APB is much higher than to form a ISF [23, 28]. Often, more than one twinning system is activated which results in extensive interaction of twins belonging to dierent twinning systems (cross twinning) [13]. It is important to note that twinning is unidirectional as opposed to dislocation slip which is bi-directional. Twins can form during solid state reactions like recrystallization and phase transformations as well as during plastic deformation [25±27]. It has been shown that twinning just as dislocation slip can contribute to plastic strain [26] and there is no doubt about mechanical twinning being part of the microstructural processes controlling creep in nearg-TiAl alloys [7, 8, 12±19]. An example of a twinned equiaxed g-grain is shown in Fig. 2. Early studies on deformation of g-TiAl at ambient and elevated temperature already reported that twins play an important role in the deformation process of this intermetallic alloy [12, 13]. Deformation twinning is observed in a wide temperature range (20±9008C) and becomes more active with increasing deformation temperature [29]. Twinning is apparently an ``easy'' deformation mode (as compared with superdislocations) which is due to the low intrinsic stacking fault energy. The energy for the formation of a complex stacking fault or an APB is much higher [24±28]. The formation of a mechanical twin requires less energy and, consequently, is more favorable than other deformation modes. Yoo [15] and Feng et al. [30] pointed out that in ordered intermetallic alloys, deformation twinning is an important micromechanical feature although it is commonly believed to be the dominant mode only at high strain rates and/or low temperatures. It appears that twinning is a mechanism which can already occur at stresses near the elastic limit without preliminary hardening. Farenc et al. [33] demonstrated that in a single-phase g-alloy, the occurrence of twinning with respect to other deformation modes can be described approximately by Schmid's law. They found that the critical resolved shear stress for twinning is higher than that of simple dislocation glide and a little lower than that of superdislocation glide. The apparent critical resolved shear stress (CRSS) for twinning was measured to be rather low, about 200 MPa at 7278C and 100 MPa at 9278C [31]. Potential nucleation sites for twinning in single-phase alloys are grain boundaries, transformation twin boundaries, dislocations, surfaces and crack tips [31]. In polycrystalline alloys, grain boundaries are the most potent sites. Deformation twins in g-TiAl are readily nucleated at interfaces such as g/g and g/a2 lamellar interfaces and grain boundaries [31]. Jin and Bieler [32] showed that mechanical twins in near-g-TiAl nucleate as bowed partial dislocations emanating from grain or interfacial boundaries where there are high
SKROTZKI: DEFORMATION TWINNING
stress concentrations. In contrast, Farenc et al. [33] found that twins are nucleated within the grain interior by a pole mechanism due to a partial dislocation turning around a perfect dislocation. The pole mechanism was generally proposed by Cottrell and Bilby [34] for the nucleation of twins in b.c.c. crystals. Their model assumes that a pole dislocation exists with a Burgers vector perpendicular to the twinning plane. A partial dislocation on the twinning plane rotates around the pole, produces twinning in successive layers and generates a homogeneous twin. According to von Mises [35], normally ®ve independent slip systems have to be activated to accommodate a certain strain increment in a crystal. Mecking et al. [21] and Goo [36] have shown that ordinary slip and twinning systems do not provide ®ve independent slip systems and that superdislocations are always needed to ful®ll the von Mises criterion. However, a number of studies revealed that the majority of dislocations found after deformation are ordinary dislocations and only a small number of superdislocations were present. The low activity of superdislocations was attributed to their pinning due to dissociation reactions [10, 14]. Hirth and Lothe [37] pointed out that twinning is an important deformation mechanism in crystals which have only a limited number of slip systems. In those structures, twinning can provide additional slip systems. This was also emphasized by Barrett and Massalski [25] who see the signi®cance of twinning for the deformation process in the fact that the
853
reoriented twinned crystal may provide slip planes that are favorably oriented for further deformation by slip which is well known from zinc. The objective of the present study was to examine and visualize the crystallographic aspects of mechanical twinning in g-TiAl. The role of the twin interface will be addressed with respect to its interaction with slip dislocations. Besides, possible slip processes in the twinned g-crystal will be evaluated. It is expected that this work will assist in answering the question of whether deformation twinning is a work hardening or a microstructural softening process. The consequences for deformation processes in the ordered intermetallic material are discussed. 2. LENGTH CHANGE BY TWINNING
In a ®rst step the length change of a g-crystal after deformation twinning is evaluated. Deformation or mechanical twinning is a true twinning mode with a 1808 rotation on the {111} plane resulting in a coherent twin interface. Pseudo twins and 1208 rotation twins are found in lamellar a2 g grains (after the g-phase precipitates form the aphase) which have semi-coherent interfaces to the parent g-lamella. However, these two twin modes were not experimentally observed after deformation of equiaxed g-grains [14, 22]. The four possible true twinning systems and the 12 h110] slip systems of the ordered L10 lattice are given in Tables 1 and 2. Figure 1(b) shows a true twin in a g-crystal with the twin plane K1 111 which stays invariant during
Fig. 2. TEM micrograph of a g-grain with mechanical twins after creep deformation T 7008C, s 255 MPa, e 2% of a Ti±47Al±2Nb±2Mn+0.8 vol.% TiB2 alloy [19].
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SKROTZKI: DEFORMATION TWINNING
Table 1. Slip systems in g-TiAl: h110] slip occurs by ordinary dislocations and h101] slip by superdislocations. Slip directions of the twinned crystal were calculated by applying equation (16) to the six possible glide dislocations of the parent g-crystal for the 111112 twinning system No. 1 2 3 4 5 6 7 8 9 10 11 12
Mode
Slip system in g
Slip direction of b in gT
h110] slip 0 0 0 h101] slip 0 0 0 0 0 0 0
11 1110 1 11110 1 1 111 0 11111 0 1 1 1011 11 1011 1 1 1101 1 11101 1 1101 1 11101 1 1111 01 11 11 01
1 1 4 1 1 4 11 0 11 0 4 1 1 4 1 1 1 4 1 1 4 1 01 1 01 1 1 01 1 01
deformation. Deformation twins are formed under an external shear stress which moves successive lattice planes into the twinning direction Z1 112 : The ®lled circles in Fig. 1(b) represent, e.g. Ti atoms, the open circles Al atoms that are above and below the drawing plane. The shaded area in Fig. 1(b) corresponds to that of Fig. 1(a). The shear vector, s, which transforms the parent crystal into the twinned crystal is 1=6112 : As a result of the twin deformation process the stacking sequence ABCABC is changed to ABCBAC. The shear is
homogeneous with a shear angle of 2j0 38:948 and, therefore, an amount of shear of 2 tan j0 0:707: On the left-hand side of Fig. 3, a 11 0 plane of an untwinned g-crystal is shown. (The atoms lying above and below the drawing plane are omitted.) Two straight dashed lines mark the (111) twin plane. The right-hand side of Fig. 3 shows the same g-crystal with the same number of atoms after a deformation twin has formed. It is clear from Fig. 3 that in contrast to ordinary dislocations (which do not have a c-component), twinning does contribute to deformation in the [001] direction, i.e. the g-crystal is shortened in the c-direction by the external shear. Matrix algebra can be used to describe the twinning process as was, e.g. shown by Schumann [26]. This mathematical treatment is in the following applied to g-TiAl to quantify the length change by deformation twinning. Due to the small tetragonality of the L10 unit cell, it seems justi®able to treat it as a f.c.c. structure. According to the general matrix for pure shear, the shear matrix, (S ), for twinning in f.c.c. is S E md0 p00
1
where E is the unit p matrix, m is the amount of shear which is 1= 2 for f.c.c., d0 is the direction of shear and p00 is the shear plane. For the twinning system 111112 (No. 8 in Table 2)
Table 2. Schmid factors, m, for ordinary slip in the parent g-crystal and in the twinned crystal for dierent applied stress directions, sappl. Directions of the twinned crystal were calculated applying the transformation matrices [e.g. equation (16)] to directions in the parent g-phase Direction of sappl No. 1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
h110] slip 0 0 0 Twinning 0 0 0 111112 h110] slip 0 0 0 1 111 12 h110] slip 0 0 0 11 111 2 h110] slip 0 0 0 1 1 11 1 2 h110] slip 0 0 0
System
[001]g
[011]g
[010]g
1 10g
1 11g
11 1110 1 11110 1 1 111 0 11111 0 1 111 12 11 111 2 1 1 11 1 2 111112
0 0 0 0 ÿ0.471 ÿ0.471 ÿ0.471 ÿ0.471 2 2 1T ÿ0.108 ÿ0.108 0 0 22 1T 0 0 0.181 0.181 2 21T 0 0 ÿ0.181 ÿ0.181 221T 0.181 0.181 0 0
0 0.408 0 ÿ0.408 ÿ0.236 0 0 ÿ0.236 4 1 1 T 0.454 ÿ0.227 ÿ0.272 0.408 41 1 T 0.272 ÿ0.408 ÿ0.454 0.227 011T 0 0.408 0 ÿ0.408 011T 0 0.408 0 ÿ0.408
ÿ0.408 0.408 0.408 ÿ0.408 0.236 0.236 0.236 0.236 2 12 T 0.227 ÿ0.045 0.136 0.408 212 T ÿ0.136 ÿ0.408 ÿ0.227 0.045 212T 0.408 0.136 ÿ0.045 0.227 2 12T 0.045 ÿ0.227 ÿ0.408 ÿ0.136
0 0 0 0 0.471 0.471 0 0 1 10T 0 0 0 0 11 4 T 0 0 ÿ0.181 ÿ0.181 11 4T 0 0 0.181 0.181 1 10T 0 0 0 0
0 0 ÿ0.272 ÿ0.272 0 0.314 ÿ0.157 ÿ0.157 5 11T 0.302 ÿ0.423 ÿ0.454 0.272 11 1 T 0 0 ÿ0.272 ÿ0.272 1 15T 0 0 ÿ0.151 0.151 1 51T ÿ0.302 0.423 0.272 ÿ0.454
11 1110 1 11110 1 1 111 0 11111 0 11 1110 1 11110 1 1 111 0 11111 0 11 1110 1 11110 1 1 111 0 11111 0 11 1110 1 11110 1 1 111 0 11111 0
SKROTZKI: DEFORMATION TWINNING
d0
d 112 p jdj 6
855
l u00 Mu0 ÿ1=2 :
2
The strain, d0, is then given by
and
d0 l ÿ 1 100% p 111 p00 p : jpj 3
3
The distortion matrix is equivalent to the inverse shear matrix: S ÿ1 E ÿ md0 p00 :
4
The deformation matrix, M, is given by 0
M S ÿ1 S ÿ1
7
5
8
and the total strain, d, is then d q d0
9
where q is the volume of the twinned crystal. For the c-direction, u0 001, the length change is 2 6ÿ l4 0 0
0 1B 5 1 @ 1 6 2
1 5 2
10 1 3ÿ1=2 0 2 C 7 10 2 A@ 0 A 5 1 11
0
where S ÿ1 is the transpose of the distortion matrix. For the 111112 twinning system of the f.c.c. lattice the deformation matrix is calculated using equations (1)±(4) 0
1B 5 M @ 1 6 2
1 5 2
1 2 C 2 A: 11
6
The change of length, l, of a crystal direction, u0, due to deformation twinning can be calculated using the following equation:
which gives l10:739 and d0 ÿ26:1%: The volume fraction, q, of twinned crystal shown in Fig. 3(b) amounts to 14 atoms divided by 70 atoms, i.e. q 0:2: This results in a total strain of d ÿ 5:2% which is in fact a contraction in the cdirection of the L10 lattice. For u0 100 and [010] positive strains of d0 9:5% are obtained. These length changes are qualitatively represented in the stereographic projection in Fig. 4(a) for the twinning system 111112 , which shows several traces of planes: K1 is the twin plane (111)g and is invariant, i.e. remains unrotated and undeformed; K2
Fig. 3. (a) Parent g-crystal cut through the 11 0 plane. Dashed lines mark the (111)-twinning plane. (b) g-crystal with 111112 deformation twin. The original g-crystal is shortened by the amount of d in the c-direction [001] and lengthened in the [110] direction. The original coordinate system of the parent crystal is transformed into a new coordinate system (index T) for the twinned crystal.
856
SKROTZKI: DEFORMATION TWINNING
Fig. 4. Stereographic projection illustrating the crystallography of twinning for the: (a) 111112 ; (b) 1 1 11 1 2 ; (c) 11 111 2 ; (d) 1 111 12 twinning system. Crystal directions in the hatched areas are shortened and the twinning process lengthens those in the other areas.
SKROTZKI: DEFORMATION TWINNING
Fig. 4 (continued)
857
858
SKROTZKI: DEFORMATION TWINNING
111 g rotates into K20 115 g 111 T but remains undeformed; K0 112 g is perpendicular to the twin plane K1. Zi is the length of the main diagonals of the deformation ellipsoids, which can be calculated from the eigenvalues of the deformation matrix, M, and are combined with its eigenvectors [26]. Zi 1 does not result in a deformation, Zi < 1 represents a shortening and Zi > 1 a lengthening (hatched region) of crystal directions, respectively. It becomes clear from Fig. 4(a) that, in contrast to the c-direction, the [100] and [010] directions are lengthened by mechanical twinning while 11 0, 1 1 2, 1 01 and 1 10 remain unchanged. The length change and the strain due to twinning can be easily calculated for twinning systems 5±7 of Table 2. The qualitative results are represented in Figs 4(b)±(d). It is important to note that deformation twinning obviously always results in a shortening of the [001] direction whereas the [100] and [010] directions are always lengthened.
tal is accomplished by 0
R1808 y3
1 B @0 0
1 0 C 0 A: 1
0 1 0
13
The inversion matrix 0
1 B Z @ 0 0
1 0 C 0A 1
0 1 0
14
provides the correct sign for the axes of the twinned crystal. The twinning process is a combination of the rotation and the inversion: 0
Z R1808 y3
1 0 @0 1 0 0
1 0 0 A: 1
15
The transformation matrix for the base g is then obtained by 3. TRANSFORMATION MATRIX FOR TWINNING
The transformation matrix can be developed following, e.g. Schumann's approach to describe the orientation relationship between the parent phase and the product phase [26]. In a ®rst step, an orthonormal coordinate system, Y, is de®ned for the gcrystal with the twinning system 111112 in Fig. 1(b) with y3 parallel to the twin axis [111]g, y2 parallel to 112 g and y1 perpendicular to both of them. The base Y is then given by 11 0g y1 p , 2
112 g y2 p , 6
111g y3 p : 3
The base Y is connected to the base g1 100, g2 010 and g3 001 by 0 1 1 1 p p B p2 6 3 B B B 1 1 1 p gTY y1 y2 y3 B B ÿ p2 p6 3 B B @ 2 1 0 ÿ p p 6 3
1 B p2 0 1 B B y1 B 1 YTg @ y2 A B B p6 B y3 B @ 1 p 3
1 ÿ p 2 1 p 6 1 p 3
g (with 1 C C C C C 12a C C C A
1 0
C C C 2 C ÿ p C : 6C C C A 1 p 3
0
R111g
1B 1 @ 2 3 2
2 1 2
1 2 C 2 A: 1
16
The transformation matrices for the three remaining twinning systems can be calculated accordingly.
11
and 0
R111g gTY Z R YTg
12b
The rotation of the g-crystal into the twinned crys-
4. INTERACTION OF GLIDE DISLOCATIONS WITH TWIN INTERFACES
Although mechanical twinning is generally regarded as an important deformation mechanism in g-TiAl, some controversy remained whether deformation twinning must be considered as a work hardening mechanism or as a microstructural softening process. Seo et al. [18] and Morris and coworkers [7, 8] argued that twin interfaces hinder dislocation motion and, therefore, harden the gphase. However, Sleeswyk and Verbraak [38] have shown for the b.c.c. crystal structure that from a crystallographic point of view the coherent twin boundary presents only a minor obstacle to slip and that all slip dislocations can pass the boundary by suitable dissociation. This question will be addressed in the following (exempli®ed for the 111112 twinning system) by applying the transformation matrix of equation (16) to the six possible glide dislocation Burgers vectors (two for ordinary and four for superdislocations) of Table 1:
0
SKROTZKI: DEFORMATION TWINNING
1
0 1 1 1@ A 1B 1 C 1 4 @ 1 A , 2 6 0 4 T
0 1 0 1 0 4 1 C @ 1 A4 B @ 1 A , 3 1 1 0
1
T
0 1 1 1 @ 0 A4@ 0 A , 1 1 T
0 1 0 1 1 1 C @ 0 A4 1B @ 4 A , 3 1 1 T
0 1 0 1 1 1 1@ A 1@ A , 1 4 1 2 2 0 0 T
17
0
1 0 1 0 0 @ 1 A4@ 1 A : 1 1 T
The results demonstrate that the last three Burgers vectors of equation (17) (i.e. 50% of the possible glide dislocations) remain unchanged in the twinned crystal. The Burgers vectors of these dislocations are lying in the (111)-twinning plane and, therefore, the interface does not represent an obstacle for the moving dislocation. The remaining Burgers vectors are transformed into 1=61 1 4 , 1=31 4 1 or
859
1=34 1 1 : It is not likely that these dislocations can pass the twin interface. Generally, the passage of a slip dislocation causes two halves of a crystal to move past each other over a distance which is equal to the Burgers vector, b. Considering the {111} twinning plane, a step in this layer of atoms will result. The component of the step height perpendicular to the {111} plane can be expressed as numbers of {111} layers exposed. The component lying within the {111} plane is of no interest for the following discussion. The height of the step depends on the magnitude and on the orientation of b. Let us consider the (111)-twinning plane. The Burgers vectors 1=211 0, 101 and 011 are lying within the {111} plane and do not cause any steps. b 1=2110 causes a step of height one while superdislocations with b 101 and [011] cause steps of height two. Taking into account the resulting step heights, the product dislocations of equation (17) can dissociate further into 1 1 1 1 1 4 T 4 1 1 0T 112 T 6 2 3 1 1 1 1 4 1 T 4 1 01 T 12 1T 12 1T 3 3 3
18
1 1 1 4 1 1 T 4 01 1 T 2 11T 2 11T : 3 3 3
Fig. 5. Possible dislocation interaction with a twin boundary. (a) Ordinary dislocations with b0 1=2110 approach to a twin interface. (b) Dissociation of an ordinary dislocation into a double Shockley partial dislocation which remains in the interface and into an ordinary dislocation of the twinned crystal which dissociates further when the second twin interface is reached (c).
A similar dissociation reaction was proposed for b.c.c. structures [37, 38]. The dissociation process of equation (18) results in an ordinary dislocation or a superdislocation of the twinned crystal and residual dislocations that remain in the twin interface [37]. Dislocations with b 1=3h112i are double Shockley partials which were predicted and experimentally observed in dissociation reactions of superdislocations [39, 40]. This reaction is illustrated in Fig. 5. In the presence of a favorable external shear stress, the twinning dislocation can let the twin grow. Hence, the twinning interface does not act as a major barrier to glide dislocations. However, the dissociations of equation (18) are energetically unfavorable and can only occur under large applied stresses. If the dissociation processes of equation (18) do not take place (e.g. because the applied stress is too low) then the twin boundary represents an obstacle to the moving dislocation and pileups will form. Further experimental TEM work is required to study this process in greater detail. Another possible dissociation reaction of the product dislocation of the ordinary dislocation of equation (17) is 1 1 1 1 1 4 T 4 1 1 1 T 112 T : 6 3 6
19
Wang et al. [41] have observed 1=3111 Frank partials at ledges of incoherent twin boundaries of
860
SKROTZKI: DEFORMATION TWINNING
lamellar microstructures. However, all those twins had the (111)-twinning plane parallel to existing lamellar interfaces. Twins formed at a2/g or g/gt interfaces exhibit coherent twin boundaries and no ledges and no Frank partials were experimentally observed [41]. Therefore, the dissociation reaction of equation (18) appears to be more likely than that of equation (19). Generally, the twin boundary represents a special case of the grain boundary. Gleiter et al. [42] and BaÈro et al. [43] have developed a model, which shows that the grain boundary plane is sheared when the glide plane of a slip dislocation changes at the grain boundary. The shear component perpendicular to the grain boundary plane causes a grain boundary ledge while the parallel shear component forms a grain boundary dislocation. The energy required to shear the grain boundary is composed of (i) the surface and (ii) the strain energy of the ledge and (iii) the self-energy to form the grain boundary dislocation. The strain of the grain boundary ledge is lowest when the slip planes of both grains intersect in the grain boundary and the shear, which has to be accommodated by the grain boundary area, is oriented in such a way that its direction is parallel to the grain boundary. This is exactly the case for the coherent twin boundary shown in Fig. 5. Now only the surface energy of the ledge and the self-energy of the twin dislocation has to be provided. Elastic strains do not arise because twin boundary dislocations accommodate them. Hence, this case is most favorable for the passage of dislocations through an interface.
Fig. 6. Qualitative representation of the four deformation twinning systems with respect to the sense and orientation of a uniaxial applied load within the extended unit triangle for the tetragonal crystal. In the crossed regions, twinning is forbidden under tensile load while the hatched regions represent orientations where twinning is impossible under compression: (a) 1 11-, (b) 11 1-, (c) 1 1 1-, and (111)twinning planes.
5. EFFECT OF ORIENTATION OF THE TWINNED g-CRYSTAL
It is clear from Fig. 3(b) that the twinned crystal has been reoriented by the twinning process and is now dierently oriented to an externally applied stress than the parent g-grain. In the upper part of Table 2, the Schmid factors, m, of the four ordinary dislocation slip systems (1±4) and of the four twinning systems (5±8) are summarized for some applied stress directions of the g-crystal. The Schmid factors of the h101] superdislocation slip systems are not shown because previous studies revealed that the majority of dislocations contributing to strain were ordinary dislocations. Schmid factors of twinning and dislocation motion have been previously calculated (e.g. by Morris and Leboeuf [8], Mecking et al. [21], and Sun et al. [29]). Nevertheless, the Schmid factors are shown again in Table 2 because they are essential for the following discussion. It is obvious from Table 2 that a g-crystal cannot deform by any of the four ordinary dislocation systems (i.e. m 0 if the external stress is applied in the [001] or the 1 10 direction. Only two slip systems are activated for the 1 11 and [011] stress directions. Table 2 also gives the Schmid factors for the four possible twinning systems in g-TiAl. Note that twinning is unidirectional and the crystal behaves dierently under tension and compression. Figure 6 qualitatively represents the active twinning systems with respect to the sense and orientation of the uniaxial load within the extended unit triangle for the tetragonal crystal system. A resembling representation was given by Sun et al. [29]. The hatched regions represent orientations where twinning cannot take place under compression while the crossed region illustrates those where twinning is forbidden under tension. For the chosen standard triangle in Fig. 6, the primary twinning system (with highest Schmid factor) in tension is 11 111 2 while in compression it is 111112 and 1 111 12 as was shown by Ref. [29]. The lower parts of Table 2 show the Schmid factors for the reoriented twinned crystal. The slip systems remain the same as in the parent g-crystals. The applied stress direction for the twinned crystal is calculated by using the transformation matrices, e.g. equation (16). The results show that in the case of sappl k001 the Schmid factors of two dislocation slip systems are m 6 0 in all of the twinned crystals, i.e. the twinned crystals can deform by ordinary dislocations while the parent crystal cannot. The total strain, etot, which is exhibited by the material, has three components etot s ed r, x eT VT ed,T r, x T
20
with ed and eT being the strain due to dislocation and twin plasticity, respectively. ed,T is the strain caused by dislocation activity within the reoriented
SKROTZKI: DEFORMATION TWINNING
twin. ed and eT depend on the dislocation density, r, and the mean free path for dislocation glide in a grain, x , and in the twin, x T : The latter is rather short compared with a grain. eT is determined by the volume fraction of twins formed during deformation. It has to be noted though that the volume fraction of twins is rather low. From TEM micrographs the average twin width of the crept condition e 2% shown in Fig. 2 was measured to be 50 nm. The area fraction of twins is approximately 5%. However, twins are present in only 25% of all the g-grains (after 2% strain) because not all of them are favorably oriented for twinning. This results in a total volume fraction of twinned material of approximately 1.3%, which has to be regarded as rather low, and it is doubtful that dislocations active within the twins contribute considerably to the overall deformation. 6. SUMMARY AND CONCLUSION
The present study examined and visualized crystallographic aspects of mechanical twinning in the ordered L10 structure of the g-phase. Several authors have shown that mechanical twinning is an important deformation mechanism in near-g-TiAl alloys. From a crystallographic point of view the signi®cance of twinning manifests itself in the following elements: 1. In contrast to ordinary dislocation plasticity, mechanical twinning does have a strain component in the c-direction of the g-crystal. Consequently, mechanical twinning reduces strain incompatibilities which arise due to the activity of mainly 1/2h110] dislocations and increases the ductility of the intermetallic material. 2. The coherent twin boundary seems to represent only a minor obstacle to slip dislocations. Dislocations, which have to pass the twin interface, can do so by suitable dissociation processes. 3. Ordinary dislocations can ®nd somewhat more favorable conditions in the reoriented twin than in the parent grain for, e.g. [001]g or 1 10g stress directions. Under such conditions the twinned crystal can deform by h110] dislocations while the parent crystal cannot. 4. Obviously, all eects associated with twinning depend on the volume fraction of twins, VT, present in the material. Alloy modi®cations, which favor twinning will result in improved room temperature ductility. 5. The total strain shown by the material is composed of (i) the strain due to dislocation (ed) and twin (eT) plasticity, respectively, and the strain caused by dislocation activity within the reoriented twin (ed,T). ed and ed,T depend on the dislocation density, r, the Burgers vector, b, and
861
the mean free path for dislocations, x , while eT is determined by the volume fraction of twins. 6. Further TEM work is required to provide experimental evidence for (2) and (3).
AcknowledgementsÐThe author wishes to express her appreciation to Professors G. Eggeler and E. Hornbogen of Ruhr-University Bochum for stimulating discussion. REFERENCES 1. Isobe, S. and Noda, T., in Structural Intermetallics 1997, ed. M. V. Nathal, et al. The Minerals, Metals & Materials Society, Warrendale, PA, 1997, pp. 427± 433. 2. Tetsui, T., in Structural Intermetallics 1997, ed. M. V. Nathal, et al. The Minerals, Metals & Materials Society, Warrendale, PA, 1997, pp. 489±493. 3. Richter, H. and Smarsly, W., Mat.-wiss. u. Werkstotech., 1997, 28, 15. 4. Kim, Y.-W. and Dimiduk, D. M., J. Metals, 1991, 43, 40. 5. Kim, Y.-W., J. Metals, 1994, 46, 30. 6. Beddoes, J., Wallace, W. and Zhao, L., Int. Mater. Rev., 1995, 40, 197. 7. Morris, M. A. and Lipe, T. I., Intermetallics, 1997, 5, 329. 8. Morris, M. A. and Leboeuf, M., J. Mater. Res., 1998, 13, 1. 9. Lu, M. and Hemker, K. J., Acta mater., 1997, 45, 3573. 10. Appel, F. and Wagner, R., Mater. Sci. Engng, 1998, R22, 187. 11. Seo, D. Y., et al., Metall. Mater. Trans. A, 1998, 29A, 89. 12. Shechtman, D., Blackburn, M. J. and Lipsitt, H. A., Metall. Trans., 1974, 5, 1373. 13. Lipsitt, H. A., Shechtman, D. and Schafrik, R. E., Metall. Trans., 1975, 6A, 1991. 14. Yamaguchi, M. and Umakoshi, Y., Prog. Mater. Sci., 1990, 14, 1. 15. Yoo, M. H., J. Mater. Res., 1989, 4, 50. 16. Couret, A., Farenc, S., Caillard, S. and Coujou, A., in Twinning in Advanced Materials, ed. M. H. Yoo and M. Wuttig. The Minerals, Metals & Materials Society, Warrendale, PA, 1994, pp. 361±374. 17. Jin, Z., Cheong, S.-W. and Bieler, T. R., in Gamma Titanium Aluminides, ed. Y.-W. Kim, R. Wagner and M. Yamaguchi. The Minerals, Metals & Materials Society, Warrendale, PA, 1995, pp. 975±983. 18. Seo, D. Y., Bieler, T. R. and Larsen, D. E., in Creep and Fracture of Engineering Materials and Structures, ed. J. C. Earthman and F. A. Mohamed. The Minerals, Metals & Materials Society, Warrendale, PA, 1997, pp. 577±586. 19. Skrotzki, B., UÈnal, M. and Eggeler, G., Scripta mater., 1998, 39, 1023. 20. Viguier, B., et al., Phil. Mag. A, 1995, 71, 1295. 21. Mecking, H., Hartig, C. and Kocks, U. F., Acta mater., 1996, 44, 1309. 22. Loiseau, A. and Lasalmonie, A., Mater. Sci. Engng, 1994, 67, 163. 23. Fu, C. L. and Yoo, M. H., Scripta mater., 1997, 37, 1453. 24. Singh, S. R. and Howe, J. M., Scripta metall. mater., 1991, 25, 485. 25. Barrett, C. S. and Massalski, T. B., Structure of Metals, 3rd edn. McGraw-Hill, New York, 1966. 26. Schumann, H., Kristall-Geometrie. VEB Leipzig, 1979.
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27. Dieter, E., Mechanical Metallurgy. McGraw-Hill, New York, 1988. 28. Hug, G., Loiseau, A. and VeyssieÁre, P., Phil. Mag., 1988, 57, 499. 29. Sun, Y. Q., Hazzledine, P. M. and Christian, J. W., Phil. Mag. A, 1993, 68, 471. 30. Feng, C. R., Michel, D. J. and Crowe, C. R., Scripta metall., 1989, 23, 241. 31. Yoo, M. H., Intermetallics, 1998, 6, 597. 32. Jin, Z. and Bieler, T. R., Phil. Mag. A, 1995, 71, 925. 33. Farenc, S., Coujou, A. and Couret, A., Phil. Mag. A, 1993, 67, 127. 34. Cottrell, A. H. and Bilby, B. A., Phil. Mag., 1951, 42, 573. 35. von Mises, R., Z. angew. Math. Mech., 1928, 8, 161. 36. Goo, E., Scripta mater., 1998, 38, 1711.
37. Hirth, J. P. and Lothe, J., Theory of Dislocation, 2nd edn. Krieger, Malabar, FL, 1992. 38. Sleeswyk, A. W. and Verbraak, C. A., Acta metall., 1961, 9, 917. 39. Hug, G., Loiseau, A. and Lasalmonie, A., Phil. Mag. A, 1986, 54, 47. 40. Inkson, B. J., Phil. Mag. A, 1998, 77, 715. 41. Wang, J. G., et al., in Structural Intermetallics 1997, ed. M. V. Nathal, et al. The Minerals, Metals & Materials Society, Warrendale, PA, 1997, pp. 119± 128. 42. Gleiter, H., Hornbogen, E. and BaÈro, G., Acta metall., 1968, 16, 1053. 43. BaÈro, G., Gleiter, G. and Hornbogen, E., Mater. Sci. Engng, 1968/69, 3, 92.
CRYSTALLOGRAPHIC ASPECTS OF DEFORMATION TWINNING AND CONSEQUENCES FOR PLASTIC DEFORMATION PROCESSES IN g-TiAi B. SKROTZKI Ruhr-University Bochum, Department of Mechanical Engineering, Institute for Materials, D-44780 Bochum, Germany (Received 6 August 1999; accepted 12 October 1999) AbstractÐMechanical twinning is an important deformation mechanism in the g-phase of near-g-TiAl alloys at room temperature as well as at high temperature. The present work examines the crystallography of twinning in the ordered L10 structure of the g-phase and quanti®es the resulting deformation for dierent crystallographic directions. In addition, the open question is addressed whether twin interfaces represent obstacles for glide dislocations. It is shown that one part of the possible glide dislocations remains completely unaected by the twin interface because their Burgers vectors lie in the twin plane. The other part of the dislocations has to change its glide directions while passing the interface, which can occur by suitable dissociation processes. Prerequisite is a suciently high applied stress. The reoriented twin may be more favorably oriented for some glide directions than its parent crystal and thus promote further dislocation plasticity. The impact of this eect on ductility depends on the volume fraction of twins. # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. ZusammenfassungÐMechanische Zwillingsbildung ist ein wichtiger Verformungsmechanismus der g-Phase von near-g-TiAl-Legierungen sowohl bei Raumtemperatur als auch bei hohen Temperaturen. Die vorliegende Arbeit beleuchtet die Kristallographie der Zwillingsbildung in der geordneten L10 Struktur der g-Phase und quanti®ziert die resultierende Verformung fuÈr verschiedene kristallographische Richtungen. DaruÈber hinaus wird die bisher oene Frage angesprochen, ob Zwillingsgrenzen Hindernisse fuÈr gleitende Versetzungen darstellen. Es wird gezeigt, daû der eine Teil der moÈglichen Gleitversetzungen durch die Zwillingsgrenze vollstaÈndig unbeein¯uût bleibt, da ihre Burgers-Vektoren in der Zwillingsebene liegen. Der andere Teil der Versetzungen muû jedoch beim Passieren der Zwillingsgrenze seine Gleitrichtung aÈndern, was durch geeignete Aufspaltungsprozesse geschehen kann. Voraussetzung dafuÈr ist, daû eine ausreichend hohe aÈuûere Spannung vorhanden ist. Der neu gebildete Zwillingskristall kann fuÈr einige Gleitrichtungen guÈnstiger orientiert sein als das Ausgangskorn. Der Ein¯uû dieses Eekts auf die DuktilitaÈt haÈngt vom Volumenanteil der Zwillinge ab. # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Intermetallic compound; Twinning; Crystallography; Interface; Dislocations
1. INTRODUCTION
Intermetallic near-g-TiAl alloys are regarded as new candidate materials for high temperature applications due to their low density combined with high strength and high Young's modulus which are retained at high temperature, and good corrosion resistance. TiAl technology has reached an advanced level and high temperature structural applications such as automotive engine valves, turbocharger rotors and turbine engine components are envisaged [1±3]. Technological aspects including structural evolution during thermomechanical processing, the eect of microstructure on mechanical properties and the in¯uence of alloying elements have been studied in great detail [4, 5]. In addition,
the creep behavior has been investigated intensively during recent years because possible applications at high temperature are often subjected to conditions where creep processes are inevitable [6±11]. Generally, the deformation behavior is of great interest due to the fact that TiAl alloys (as most intermetallic alloys) do not show much ductility at room temperature. Several studies have shown that beside dislocation glide and climb plasticity, mechanical twinning is an important deformation mechanism at room temperature as well as at high temperature for many near-g-TiAl alloys [7, 8, 12± 18]. The ordered L10 structure of g-TiAl is a CuAuItype superlattice with a tetragonally distorted f.c.c.
1359-6454/00/$20.00 # 2000 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 9 9 ) 0 0 3 8 5 - 7
852
SKROTZKI: DEFORMATION TWINNING
Fig. 1. (a) Tetragonally distorted f.c.c. unit cell (L10 superlattice) of g-TiAl. (b) True twin in a g-crystal with twinning plane K1 111 and shear direction Z1 112 : Filled symbols represent Ti atoms; open symbols are Al atoms situated above and below the drawing plane.
unit cell c=a 1:02 where the Ti and Al planes alternate in the c-direction [Fig. 1(a)]. The lattice shows large similarities to the f.c.c. structure. Thus, slip occurs exclusively on {111} planes either by dislocations or by twinning. Dislocation slip is governed by ordinary (perfect) dislocations with Burgers vector b0 1=2h110{ and by superdislocations with bs1 h101 and bs2 1=2h112 [14, 20]. Ordinary dislocations do not aect the order whereas slip along h011] moves atoms to antisites and two consecutive dislocations are required to maintain the order. Deformation twins are observed but only the four twin directions of h112 -type preserve the order structure [21]. These twins are generated by Shockley partial dislocations of type bp 1=6h112 and they produce no wrong ®rst neighbors. An intrinsic stacking fault (ISF) is created in the L10 structure and there exists only one of this type per octahedral plane. These so-called order twins are the only twin modes which were experimentally found [14, 22]. The other two Shockley partial dislocations of a (111) plane produce complex stacking faults (CSF) and the twin contains
{ Mixed notations hhkl ] are used to account for the tetragonality of the L10 lattice.
wrong ®rst neighbors on each plane. The energy to form a CSF or an APB is much higher than to form a ISF [23, 28]. Often, more than one twinning system is activated which results in extensive interaction of twins belonging to dierent twinning systems (cross twinning) [13]. It is important to note that twinning is unidirectional as opposed to dislocation slip which is bi-directional. Twins can form during solid state reactions like recrystallization and phase transformations as well as during plastic deformation [25±27]. It has been shown that twinning just as dislocation slip can contribute to plastic strain [26] and there is no doubt about mechanical twinning being part of the microstructural processes controlling creep in nearg-TiAl alloys [7, 8, 12±19]. An example of a twinned equiaxed g-grain is shown in Fig. 2. Early studies on deformation of g-TiAl at ambient and elevated temperature already reported that twins play an important role in the deformation process of this intermetallic alloy [12, 13]. Deformation twinning is observed in a wide temperature range (20±9008C) and becomes more active with increasing deformation temperature [29]. Twinning is apparently an ``easy'' deformation mode (as compared with superdislocations) which is due to the low intrinsic stacking fault energy. The energy for the formation of a complex stacking fault or an APB is much higher [24±28]. The formation of a mechanical twin requires less energy and, consequently, is more favorable than other deformation modes. Yoo [15] and Feng et al. [30] pointed out that in ordered intermetallic alloys, deformation twinning is an important micromechanical feature although it is commonly believed to be the dominant mode only at high strain rates and/or low temperatures. It appears that twinning is a mechanism which can already occur at stresses near the elastic limit without preliminary hardening. Farenc et al. [33] demonstrated that in a single-phase g-alloy, the occurrence of twinning with respect to other deformation modes can be described approximately by Schmid's law. They found that the critical resolved shear stress for twinning is higher than that of simple dislocation glide and a little lower than that of superdislocation glide. The apparent critical resolved shear stress (CRSS) for twinning was measured to be rather low, about 200 MPa at 7278C and 100 MPa at 9278C [31]. Potential nucleation sites for twinning in single-phase alloys are grain boundaries, transformation twin boundaries, dislocations, surfaces and crack tips [31]. In polycrystalline alloys, grain boundaries are the most potent sites. Deformation twins in g-TiAl are readily nucleated at interfaces such as g/g and g/a2 lamellar interfaces and grain boundaries [31]. Jin and Bieler [32] showed that mechanical twins in near-g-TiAl nucleate as bowed partial dislocations emanating from grain or interfacial boundaries where there are high
SKROTZKI: DEFORMATION TWINNING
stress concentrations. In contrast, Farenc et al. [33] found that twins are nucleated within the grain interior by a pole mechanism due to a partial dislocation turning around a perfect dislocation. The pole mechanism was generally proposed by Cottrell and Bilby [34] for the nucleation of twins in b.c.c. crystals. Their model assumes that a pole dislocation exists with a Burgers vector perpendicular to the twinning plane. A partial dislocation on the twinning plane rotates around the pole, produces twinning in successive layers and generates a homogeneous twin. According to von Mises [35], normally ®ve independent slip systems have to be activated to accommodate a certain strain increment in a crystal. Mecking et al. [21] and Goo [36] have shown that ordinary slip and twinning systems do not provide ®ve independent slip systems and that superdislocations are always needed to ful®ll the von Mises criterion. However, a number of studies revealed that the majority of dislocations found after deformation are ordinary dislocations and only a small number of superdislocations were present. The low activity of superdislocations was attributed to their pinning due to dissociation reactions [10, 14]. Hirth and Lothe [37] pointed out that twinning is an important deformation mechanism in crystals which have only a limited number of slip systems. In those structures, twinning can provide additional slip systems. This was also emphasized by Barrett and Massalski [25] who see the signi®cance of twinning for the deformation process in the fact that the
853
reoriented twinned crystal may provide slip planes that are favorably oriented for further deformation by slip which is well known from zinc. The objective of the present study was to examine and visualize the crystallographic aspects of mechanical twinning in g-TiAl. The role of the twin interface will be addressed with respect to its interaction with slip dislocations. Besides, possible slip processes in the twinned g-crystal will be evaluated. It is expected that this work will assist in answering the question of whether deformation twinning is a work hardening or a microstructural softening process. The consequences for deformation processes in the ordered intermetallic material are discussed. 2. LENGTH CHANGE BY TWINNING
In a ®rst step the length change of a g-crystal after deformation twinning is evaluated. Deformation or mechanical twinning is a true twinning mode with a 1808 rotation on the {111} plane resulting in a coherent twin interface. Pseudo twins and 1208 rotation twins are found in lamellar a2 g grains (after the g-phase precipitates form the aphase) which have semi-coherent interfaces to the parent g-lamella. However, these two twin modes were not experimentally observed after deformation of equiaxed g-grains [14, 22]. The four possible true twinning systems and the 12 h110] slip systems of the ordered L10 lattice are given in Tables 1 and 2. Figure 1(b) shows a true twin in a g-crystal with the twin plane K1 111 which stays invariant during
Fig. 2. TEM micrograph of a g-grain with mechanical twins after creep deformation T 7008C, s 255 MPa, e 2% of a Ti±47Al±2Nb±2Mn+0.8 vol.% TiB2 alloy [19].
854
SKROTZKI: DEFORMATION TWINNING
Table 1. Slip systems in g-TiAl: h110] slip occurs by ordinary dislocations and h101] slip by superdislocations. Slip directions of the twinned crystal were calculated by applying equation (16) to the six possible glide dislocations of the parent g-crystal for the 111112 twinning system No. 1 2 3 4 5 6 7 8 9 10 11 12
Mode
Slip system in g
Slip direction of b in gT
h110] slip 0 0 0 h101] slip 0 0 0 0 0 0 0
11 1110 1 11110 1 1 111 0 11111 0 1 1 1011 11 1011 1 1 1101 1 11101 1 1101 1 11101 1 1111 01 11 11 01
1 1 4 1 1 4 11 0 11 0 4 1 1 4 1 1 1 4 1 1 4 1 01 1 01 1 1 01 1 01
deformation. Deformation twins are formed under an external shear stress which moves successive lattice planes into the twinning direction Z1 112 : The ®lled circles in Fig. 1(b) represent, e.g. Ti atoms, the open circles Al atoms that are above and below the drawing plane. The shaded area in Fig. 1(b) corresponds to that of Fig. 1(a). The shear vector, s, which transforms the parent crystal into the twinned crystal is 1=6112 : As a result of the twin deformation process the stacking sequence ABCABC is changed to ABCBAC. The shear is
homogeneous with a shear angle of 2j0 38:948 and, therefore, an amount of shear of 2 tan j0 0:707: On the left-hand side of Fig. 3, a 11 0 plane of an untwinned g-crystal is shown. (The atoms lying above and below the drawing plane are omitted.) Two straight dashed lines mark the (111) twin plane. The right-hand side of Fig. 3 shows the same g-crystal with the same number of atoms after a deformation twin has formed. It is clear from Fig. 3 that in contrast to ordinary dislocations (which do not have a c-component), twinning does contribute to deformation in the [001] direction, i.e. the g-crystal is shortened in the c-direction by the external shear. Matrix algebra can be used to describe the twinning process as was, e.g. shown by Schumann [26]. This mathematical treatment is in the following applied to g-TiAl to quantify the length change by deformation twinning. Due to the small tetragonality of the L10 unit cell, it seems justi®able to treat it as a f.c.c. structure. According to the general matrix for pure shear, the shear matrix, (S ), for twinning in f.c.c. is S E md0 p00
1
where E is the unit p matrix, m is the amount of shear which is 1= 2 for f.c.c., d0 is the direction of shear and p00 is the shear plane. For the twinning system 111112 (No. 8 in Table 2)
Table 2. Schmid factors, m, for ordinary slip in the parent g-crystal and in the twinned crystal for dierent applied stress directions, sappl. Directions of the twinned crystal were calculated applying the transformation matrices [e.g. equation (16)] to directions in the parent g-phase Direction of sappl No. 1 2 3 4 5 6 7 8 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
h110] slip 0 0 0 Twinning 0 0 0 111112 h110] slip 0 0 0 1 111 12 h110] slip 0 0 0 11 111 2 h110] slip 0 0 0 1 1 11 1 2 h110] slip 0 0 0
System
[001]g
[011]g
[010]g
1 10g
1 11g
11 1110 1 11110 1 1 111 0 11111 0 1 111 12 11 111 2 1 1 11 1 2 111112
0 0 0 0 ÿ0.471 ÿ0.471 ÿ0.471 ÿ0.471 2 2 1T ÿ0.108 ÿ0.108 0 0 22 1T 0 0 0.181 0.181 2 21T 0 0 ÿ0.181 ÿ0.181 221T 0.181 0.181 0 0
0 0.408 0 ÿ0.408 ÿ0.236 0 0 ÿ0.236 4 1 1 T 0.454 ÿ0.227 ÿ0.272 0.408 41 1 T 0.272 ÿ0.408 ÿ0.454 0.227 011T 0 0.408 0 ÿ0.408 011T 0 0.408 0 ÿ0.408
ÿ0.408 0.408 0.408 ÿ0.408 0.236 0.236 0.236 0.236 2 12 T 0.227 ÿ0.045 0.136 0.408 212 T ÿ0.136 ÿ0.408 ÿ0.227 0.045 212T 0.408 0.136 ÿ0.045 0.227 2 12T 0.045 ÿ0.227 ÿ0.408 ÿ0.136
0 0 0 0 0.471 0.471 0 0 1 10T 0 0 0 0 11 4 T 0 0 ÿ0.181 ÿ0.181 11 4T 0 0 0.181 0.181 1 10T 0 0 0 0
0 0 ÿ0.272 ÿ0.272 0 0.314 ÿ0.157 ÿ0.157 5 11T 0.302 ÿ0.423 ÿ0.454 0.272 11 1 T 0 0 ÿ0.272 ÿ0.272 1 15T 0 0 ÿ0.151 0.151 1 51T ÿ0.302 0.423 0.272 ÿ0.454
11 1110 1 11110 1 1 111 0 11111 0 11 1110 1 11110 1 1 111 0 11111 0 11 1110 1 11110 1 1 111 0 11111 0 11 1110 1 11110 1 1 111 0 11111 0
SKROTZKI: DEFORMATION TWINNING
d0
d 112 p jdj 6
855
l u00 Mu0 ÿ1=2 :
2
The strain, d0, is then given by
and
d0 l ÿ 1 100% p 111 p00 p : jpj 3
3
The distortion matrix is equivalent to the inverse shear matrix: S ÿ1 E ÿ md0 p00 :
4
The deformation matrix, M, is given by 0
M S ÿ1 S ÿ1
7
5
8
and the total strain, d, is then d q d0
9
where q is the volume of the twinned crystal. For the c-direction, u0 001, the length change is 2 6ÿ l4 0 0
0 1B 5 1 @ 1 6 2
1 5 2
10 1 3ÿ1=2 0 2 C 7 10 2 A@ 0 A 5 1 11
0
where S ÿ1 is the transpose of the distortion matrix. For the 111112 twinning system of the f.c.c. lattice the deformation matrix is calculated using equations (1)±(4) 0
1B 5 M @ 1 6 2
1 5 2
1 2 C 2 A: 11
6
The change of length, l, of a crystal direction, u0, due to deformation twinning can be calculated using the following equation:
which gives l10:739 and d0 ÿ26:1%: The volume fraction, q, of twinned crystal shown in Fig. 3(b) amounts to 14 atoms divided by 70 atoms, i.e. q 0:2: This results in a total strain of d ÿ 5:2% which is in fact a contraction in the cdirection of the L10 lattice. For u0 100 and [010] positive strains of d0 9:5% are obtained. These length changes are qualitatively represented in the stereographic projection in Fig. 4(a) for the twinning system 111112 , which shows several traces of planes: K1 is the twin plane (111)g and is invariant, i.e. remains unrotated and undeformed; K2
Fig. 3. (a) Parent g-crystal cut through the 11 0 plane. Dashed lines mark the (111)-twinning plane. (b) g-crystal with 111112 deformation twin. The original g-crystal is shortened by the amount of d in the c-direction [001] and lengthened in the [110] direction. The original coordinate system of the parent crystal is transformed into a new coordinate system (index T) for the twinned crystal.
856
SKROTZKI: DEFORMATION TWINNING
Fig. 4. Stereographic projection illustrating the crystallography of twinning for the: (a) 111112 ; (b) 1 1 11 1 2 ; (c) 11 111 2 ; (d) 1 111 12 twinning system. Crystal directions in the hatched areas are shortened and the twinning process lengthens those in the other areas.
SKROTZKI: DEFORMATION TWINNING
Fig. 4 (continued)
857
858
SKROTZKI: DEFORMATION TWINNING
111 g rotates into K20 115 g 111 T but remains undeformed; K0 112 g is perpendicular to the twin plane K1. Zi is the length of the main diagonals of the deformation ellipsoids, which can be calculated from the eigenvalues of the deformation matrix, M, and are combined with its eigenvectors [26]. Zi 1 does not result in a deformation, Zi < 1 represents a shortening and Zi > 1 a lengthening (hatched region) of crystal directions, respectively. It becomes clear from Fig. 4(a) that, in contrast to the c-direction, the [100] and [010] directions are lengthened by mechanical twinning while 11 0, 1 1 2, 1 01 and 1 10 remain unchanged. The length change and the strain due to twinning can be easily calculated for twinning systems 5±7 of Table 2. The qualitative results are represented in Figs 4(b)±(d). It is important to note that deformation twinning obviously always results in a shortening of the [001] direction whereas the [100] and [010] directions are always lengthened.
tal is accomplished by 0
R1808 y3
1 B @0 0
1 0 C 0 A: 1
0 1 0
13
The inversion matrix 0
1 B Z @ 0 0
1 0 C 0A 1
0 1 0
14
provides the correct sign for the axes of the twinned crystal. The twinning process is a combination of the rotation and the inversion: 0
Z R1808 y3
1 0 @0 1 0 0
1 0 0 A: 1
15
The transformation matrix for the base g is then obtained by 3. TRANSFORMATION MATRIX FOR TWINNING
The transformation matrix can be developed following, e.g. Schumann's approach to describe the orientation relationship between the parent phase and the product phase [26]. In a ®rst step, an orthonormal coordinate system, Y, is de®ned for the gcrystal with the twinning system 111112 in Fig. 1(b) with y3 parallel to the twin axis [111]g, y2 parallel to 112 g and y1 perpendicular to both of them. The base Y is then given by 11 0g y1 p , 2
112 g y2 p , 6
111g y3 p : 3
The base Y is connected to the base g1 100, g2 010 and g3 001 by 0 1 1 1 p p B p2 6 3 B B B 1 1 1 p gTY y1 y2 y3 B B ÿ p2 p6 3 B B @ 2 1 0 ÿ p p 6 3
1 B p2 0 1 B B y1 B 1 YTg @ y2 A B B p6 B y3 B @ 1 p 3
1 ÿ p 2 1 p 6 1 p 3
g (with 1 C C C C C 12a C C C A
1 0
C C C 2 C ÿ p C : 6C C C A 1 p 3
0
R111g
1B 1 @ 2 3 2
2 1 2
1 2 C 2 A: 1
16
The transformation matrices for the three remaining twinning systems can be calculated accordingly.
11
and 0
R111g gTY Z R YTg
12b
The rotation of the g-crystal into the twinned crys-
4. INTERACTION OF GLIDE DISLOCATIONS WITH TWIN INTERFACES
Although mechanical twinning is generally regarded as an important deformation mechanism in g-TiAl, some controversy remained whether deformation twinning must be considered as a work hardening mechanism or as a microstructural softening process. Seo et al. [18] and Morris and coworkers [7, 8] argued that twin interfaces hinder dislocation motion and, therefore, harden the gphase. However, Sleeswyk and Verbraak [38] have shown for the b.c.c. crystal structure that from a crystallographic point of view the coherent twin boundary presents only a minor obstacle to slip and that all slip dislocations can pass the boundary by suitable dissociation. This question will be addressed in the following (exempli®ed for the 111112 twinning system) by applying the transformation matrix of equation (16) to the six possible glide dislocation Burgers vectors (two for ordinary and four for superdislocations) of Table 1:
0
SKROTZKI: DEFORMATION TWINNING
1
0 1 1 1@ A 1B 1 C 1 4 @ 1 A , 2 6 0 4 T
0 1 0 1 0 4 1 C @ 1 A4 B @ 1 A , 3 1 1 0
1
T
0 1 1 1 @ 0 A4@ 0 A , 1 1 T
0 1 0 1 1 1 C @ 0 A4 1B @ 4 A , 3 1 1 T
0 1 0 1 1 1 1@ A 1@ A , 1 4 1 2 2 0 0 T
17
0
1 0 1 0 0 @ 1 A4@ 1 A : 1 1 T
The results demonstrate that the last three Burgers vectors of equation (17) (i.e. 50% of the possible glide dislocations) remain unchanged in the twinned crystal. The Burgers vectors of these dislocations are lying in the (111)-twinning plane and, therefore, the interface does not represent an obstacle for the moving dislocation. The remaining Burgers vectors are transformed into 1=61 1 4 , 1=31 4 1 or
859
1=34 1 1 : It is not likely that these dislocations can pass the twin interface. Generally, the passage of a slip dislocation causes two halves of a crystal to move past each other over a distance which is equal to the Burgers vector, b. Considering the {111} twinning plane, a step in this layer of atoms will result. The component of the step height perpendicular to the {111} plane can be expressed as numbers of {111} layers exposed. The component lying within the {111} plane is of no interest for the following discussion. The height of the step depends on the magnitude and on the orientation of b. Let us consider the (111)-twinning plane. The Burgers vectors 1=211 0, 101 and 011 are lying within the {111} plane and do not cause any steps. b 1=2110 causes a step of height one while superdislocations with b 101 and [011] cause steps of height two. Taking into account the resulting step heights, the product dislocations of equation (17) can dissociate further into 1 1 1 1 1 4 T 4 1 1 0T 112 T 6 2 3 1 1 1 1 4 1 T 4 1 01 T 12 1T 12 1T 3 3 3
18
1 1 1 4 1 1 T 4 01 1 T 2 11T 2 11T : 3 3 3
Fig. 5. Possible dislocation interaction with a twin boundary. (a) Ordinary dislocations with b0 1=2110 approach to a twin interface. (b) Dissociation of an ordinary dislocation into a double Shockley partial dislocation which remains in the interface and into an ordinary dislocation of the twinned crystal which dissociates further when the second twin interface is reached (c).
A similar dissociation reaction was proposed for b.c.c. structures [37, 38]. The dissociation process of equation (18) results in an ordinary dislocation or a superdislocation of the twinned crystal and residual dislocations that remain in the twin interface [37]. Dislocations with b 1=3h112i are double Shockley partials which were predicted and experimentally observed in dissociation reactions of superdislocations [39, 40]. This reaction is illustrated in Fig. 5. In the presence of a favorable external shear stress, the twinning dislocation can let the twin grow. Hence, the twinning interface does not act as a major barrier to glide dislocations. However, the dissociations of equation (18) are energetically unfavorable and can only occur under large applied stresses. If the dissociation processes of equation (18) do not take place (e.g. because the applied stress is too low) then the twin boundary represents an obstacle to the moving dislocation and pileups will form. Further experimental TEM work is required to study this process in greater detail. Another possible dissociation reaction of the product dislocation of the ordinary dislocation of equation (17) is 1 1 1 1 1 4 T 4 1 1 1 T 112 T : 6 3 6
19
Wang et al. [41] have observed 1=3111 Frank partials at ledges of incoherent twin boundaries of
860
SKROTZKI: DEFORMATION TWINNING
lamellar microstructures. However, all those twins had the (111)-twinning plane parallel to existing lamellar interfaces. Twins formed at a2/g or g/gt interfaces exhibit coherent twin boundaries and no ledges and no Frank partials were experimentally observed [41]. Therefore, the dissociation reaction of equation (18) appears to be more likely than that of equation (19). Generally, the twin boundary represents a special case of the grain boundary. Gleiter et al. [42] and BaÈro et al. [43] have developed a model, which shows that the grain boundary plane is sheared when the glide plane of a slip dislocation changes at the grain boundary. The shear component perpendicular to the grain boundary plane causes a grain boundary ledge while the parallel shear component forms a grain boundary dislocation. The energy required to shear the grain boundary is composed of (i) the surface and (ii) the strain energy of the ledge and (iii) the self-energy to form the grain boundary dislocation. The strain of the grain boundary ledge is lowest when the slip planes of both grains intersect in the grain boundary and the shear, which has to be accommodated by the grain boundary area, is oriented in such a way that its direction is parallel to the grain boundary. This is exactly the case for the coherent twin boundary shown in Fig. 5. Now only the surface energy of the ledge and the self-energy of the twin dislocation has to be provided. Elastic strains do not arise because twin boundary dislocations accommodate them. Hence, this case is most favorable for the passage of dislocations through an interface.
Fig. 6. Qualitative representation of the four deformation twinning systems with respect to the sense and orientation of a uniaxial applied load within the extended unit triangle for the tetragonal crystal. In the crossed regions, twinning is forbidden under tensile load while the hatched regions represent orientations where twinning is impossible under compression: (a) 1 11-, (b) 11 1-, (c) 1 1 1-, and (111)twinning planes.
5. EFFECT OF ORIENTATION OF THE TWINNED g-CRYSTAL
It is clear from Fig. 3(b) that the twinned crystal has been reoriented by the twinning process and is now dierently oriented to an externally applied stress than the parent g-grain. In the upper part of Table 2, the Schmid factors, m, of the four ordinary dislocation slip systems (1±4) and of the four twinning systems (5±8) are summarized for some applied stress directions of the g-crystal. The Schmid factors of the h101] superdislocation slip systems are not shown because previous studies revealed that the majority of dislocations contributing to strain were ordinary dislocations. Schmid factors of twinning and dislocation motion have been previously calculated (e.g. by Morris and Leboeuf [8], Mecking et al. [21], and Sun et al. [29]). Nevertheless, the Schmid factors are shown again in Table 2 because they are essential for the following discussion. It is obvious from Table 2 that a g-crystal cannot deform by any of the four ordinary dislocation systems (i.e. m 0 if the external stress is applied in the [001] or the 1 10 direction. Only two slip systems are activated for the 1 11 and [011] stress directions. Table 2 also gives the Schmid factors for the four possible twinning systems in g-TiAl. Note that twinning is unidirectional and the crystal behaves dierently under tension and compression. Figure 6 qualitatively represents the active twinning systems with respect to the sense and orientation of the uniaxial load within the extended unit triangle for the tetragonal crystal system. A resembling representation was given by Sun et al. [29]. The hatched regions represent orientations where twinning cannot take place under compression while the crossed region illustrates those where twinning is forbidden under tension. For the chosen standard triangle in Fig. 6, the primary twinning system (with highest Schmid factor) in tension is 11 111 2 while in compression it is 111112 and 1 111 12 as was shown by Ref. [29]. The lower parts of Table 2 show the Schmid factors for the reoriented twinned crystal. The slip systems remain the same as in the parent g-crystals. The applied stress direction for the twinned crystal is calculated by using the transformation matrices, e.g. equation (16). The results show that in the case of sappl k001 the Schmid factors of two dislocation slip systems are m 6 0 in all of the twinned crystals, i.e. the twinned crystals can deform by ordinary dislocations while the parent crystal cannot. The total strain, etot, which is exhibited by the material, has three components etot s ed r, x eT VT ed,T r, x T
20
with ed and eT being the strain due to dislocation and twin plasticity, respectively. ed,T is the strain caused by dislocation activity within the reoriented
SKROTZKI: DEFORMATION TWINNING
twin. ed and eT depend on the dislocation density, r, and the mean free path for dislocation glide in a grain, x , and in the twin, x T : The latter is rather short compared with a grain. eT is determined by the volume fraction of twins formed during deformation. It has to be noted though that the volume fraction of twins is rather low. From TEM micrographs the average twin width of the crept condition e 2% shown in Fig. 2 was measured to be 50 nm. The area fraction of twins is approximately 5%. However, twins are present in only 25% of all the g-grains (after 2% strain) because not all of them are favorably oriented for twinning. This results in a total volume fraction of twinned material of approximately 1.3%, which has to be regarded as rather low, and it is doubtful that dislocations active within the twins contribute considerably to the overall deformation. 6. SUMMARY AND CONCLUSION
The present study examined and visualized crystallographic aspects of mechanical twinning in the ordered L10 structure of the g-phase. Several authors have shown that mechanical twinning is an important deformation mechanism in near-g-TiAl alloys. From a crystallographic point of view the signi®cance of twinning manifests itself in the following elements: 1. In contrast to ordinary dislocation plasticity, mechanical twinning does have a strain component in the c-direction of the g-crystal. Consequently, mechanical twinning reduces strain incompatibilities which arise due to the activity of mainly 1/2h110] dislocations and increases the ductility of the intermetallic material. 2. The coherent twin boundary seems to represent only a minor obstacle to slip dislocations. Dislocations, which have to pass the twin interface, can do so by suitable dissociation processes. 3. Ordinary dislocations can ®nd somewhat more favorable conditions in the reoriented twin than in the parent grain for, e.g. [001]g or 1 10g stress directions. Under such conditions the twinned crystal can deform by h110] dislocations while the parent crystal cannot. 4. Obviously, all eects associated with twinning depend on the volume fraction of twins, VT, present in the material. Alloy modi®cations, which favor twinning will result in improved room temperature ductility. 5. The total strain shown by the material is composed of (i) the strain due to dislocation (ed) and twin (eT) plasticity, respectively, and the strain caused by dislocation activity within the reoriented twin (ed,T). ed and ed,T depend on the dislocation density, r, the Burgers vector, b, and
861
the mean free path for dislocations, x , while eT is determined by the volume fraction of twins. 6. Further TEM work is required to provide experimental evidence for (2) and (3).
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