Plastic deformation of bicrystals composed of polysynthetically twinned TiAl crystals

Acta mater. 49 (2001) 1009–1019 www.elsevier.com/locate/actamat

PLASTIC DEFORMATION OF BICRYSTALS COMPOSED OF POLYSYNTHETICALLY TWINNED TiAl CRYSTALS V. PAIDAR1†, D. IMAMURA2, H. INUI2 and M. YAMAGUCHI2 1

2

Institute of Physics, Academy of Sciences, Na Slovance 2, 182 21 Praha 8, Czech Republic and Department of Materials Science and Engineering, Kyoto University, Sakyo-ku, Kyoto 606-8501, Japan ( Received 6 September 2000; received in revised form 17 November 2000; accepted 17 November 2000 )

Abstract—Two types of interfaces can be distinguished in TiAl polycrystals: special rotational interfaces on the {111} planes for which the misorientation angles about the normal to the interface plane are multiples of 60°; and grain boundaries in common sense with arbitrary orientations of the boundary plane and rotation axis. The former are the boundaries in the lamellar structure parallel to one of the close packed atomic planes and the latter separate differently oriented colonies of such lamellae. The effect of both these interfaces on the plastic deformation of TiAl intermetallics is discussed. In particular, the lamellar interfaces as strong obstacles for moving dislocations induce a specific type of combined plastic deformation when the glide of ordinary dislocations operates simultaneously with twinning. On the other hand, the grain boundary can affect the dislocation processes at its vicinity by the arising compatibility stresses.  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Intermetallic compounds; Mechanical properties (plastic); Microstructure; Grain boundaries

1. INTRODUCTION

Great attention has been paid in recent years to twophase TiAl alloys because of their high specific mechanical strength and good oxidation resistance at high temperatures. It was found that contrary to common behaviour of intermetallics at low temperatures, the two-phase mixtures of g- and a2-phases are less brittle than one-phase alloys. A possible explanation could be deduced from a very particular combination of deformation modes that operate simultaneously in the lamellar structure of these ordered alloys. The abovementioned properties make these alloys very attractive candidates for applications in aerospace and automotive industries. The microstructure of titanium-rich TiAl alloys is composed of lamellae of the g- and a2-phases which are parallel to the closed packed atomic planes, that is, to the octahedral (111) and basal (0001) planes in the nearly cubic and hexagonal lattices, respectively. For the titanium-rich compositions close to the stoichiometric one, the a2-phase lamellae are narrow and not much frequent. Therefore, the majority of lamellar interfaces is of the g–g type. Each colony composed of a single set of aligned lamellae represents a grain. The samples containing just one orientation of lamel-

† To whom all correspondence should be addressed. E-mail address: [email protected] (V. Paidar)

lae can be prepared by the floating zone method and are called polysynthetically twinned (PST) crystals as they include a large number of twin interfaces [1, 2]. The twist boundaries separating the neighbouring lamellae are special rotational interfaces on the {111} planes for which the misorientation angle about the axis perpendicular to the interface plane is a multiple of 60°. On the other hand, the boundaries separating the lamellar colonies can have arbitrary misorientation, and also their plane is not fixed to a certain crystallographic orientation. Those are in fact ordinary grain boundaries in a common sense of their usage in the literature. The superlattice of the g-phase is of the L10 type that is tetragonal but the parent lattice of the ordered alloy is only slightly different from the fcc lattice. This is why the cubic notation with the mixed parentheses {hkl) and 具uvw] is usually used. It differentiates the first two indices from the non-equivalent third one of the tetragonal structure. Three types of lamellar interfaces can occur between the lamellae of the L10 type. These interfaces are twist grain boundaries with the special rotation angles of 60, 120 and 180° about the [111] axis normal to the interface plane and are called pseudotwin, rotational and truetwin interfaces, respectively. The superlattice of the a2-phase is hexagonal and of the D019 type. In both phases the lamellae are parallel to the (111) or (0001) close packed planes. It was found by quantitative

1359-6454/01/$20.00  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 0 0 7 - 6

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PAIDAR et al.: PLASTIC DEFORMATION OF BICRYSTALS

study of the TiAl microstructure that the occurrence of two possible stacking sequences of the fcc lattice (ABC or CBA) in the L10 structure are equally probable [3]. This is apparently related to the fact that the sequences in the opposite directions induce the strain fields of the opposite sign during the formation of the g-phase by the transformation from the high temperature hexagonal phase. In general, six different rotational variants of the g-phase have to be distinguished in each grain. Three of them are associated with three different cubic axes, namely [100], [010] and [001], multiplied by two due to two different atomic stackings parallel to the (111) interface plane, direct and inverse one. The grains in the cast samples are usually equiaxed but can form columnar structure when directionally solidified. Such structures deserve special attention because of their better mechanical properties. The grains are elongated in the direction of the growth axis and hence the grain boundary normals are perpendicular to the growth direction. This paper will focus on the mono- and bicrystals loaded along the 具112] growth axis, so-called A2 orientation [4–6] for which the most detailed experimental observations are available. The mechanical properties of PST crystals are obviously strongly anisotropic and depend on the orientation of the loading axis with respect to the normal to the lamellar interfaces [5, 7, 8]. With the exception of the loading axis normal to the lamellae, the plastic deformation takes place parallel to the lamellar interfaces. This behaviour is natural for the inclination of the loading axis close to 45°, as the slip plane parallel to the interfaces experiences the largest Schmid factor. However, it is rather unexpected for the loading axis parallel to the interfaces, when the Schmid factor on the (111) plane parallel to the interfaces is zero and the plastic deformation takes place on the {111} planes inclined to the lamellae. It can be anticipated that the presence of interfaces as sufficiently strong obstacles for dislocation motion induces a special channelling of shear strain along the interfaces as discussed below. The effect of the grain boundaries between the colonies of aligned lamellae on plastic deformation can be treated on different levels: the additional stresses arise in the vicinity of the boundary due to compatibility requirements on the macroscopic level of elastic continuum as will be discussed in the next section. The boundary structure including the steps and ledges, and dislocation reactions leading to the formation of residual boundary dislocations, can be taken as being on a mesoscopic level. Finally, the deformation processes can be affected on the atomic level by the boundary structure and chemistry, segregation phenomena etc. In this paper, we will concentrate on the grain boundaries between the PST crystals, and the lamellar interfaces as special twist boundaries will be discussed only for comparison purposes. First we briefly introduce a model for the evaluation of the compati-

bility stresses, and then the plastic deformation of TiAl mono- and bi-crystals will be described and compared with theoretical predictions. 2. COMPATIBILITY STRESSES

The PST crystal is composed of different variants of the g-phase with the tetragonal symmetry. However, the overall average symmetry is hexagonal with the c-axis parallel to the normal to the {111} interfaces. Obviously, the occurrence of the a2 phase, that has the hexagonal symmetry itself, does not change this symmetry. Therefore, the hexagonal symmetry will be considered in the calculations of compatibility stresses on the macroscopic level of elastic continuum. Nevertheless, since the properties of the 具101典 dislocations depend on the orientation of the tetragonal axis in each variant, the tetragonality will be respected on the mesoscopic level of the forces acting on the dislocations. Let us discuss the bicrystals composed of two joint PST crystals depicted in Fig. 1. As mentioned above, contrary to the lamellar interfaces, the grain boundaries between PST crystals need not be of special character since their misorientation can be in principle arbitrary. Nevertheless, we will consider only the bicrystals with the normal to the grain boundary perpendicular to the [1¯ 1¯ 2] growth axis. When the (111) lamellar interfaces are parallel to [1¯ 1¯ 2], there are still two angles characterizing the geometry of the bicrystal, namely, the misorientation angle between the two PST crystals and the inclination of the boundary plane. We can reduce them to just one variable, if we consider only the lamellae in grain B parallel to the

Fig. 1. Bicrystal composed of grains A and B representing two lamellar colonies. q is the misorientation of PST crystals.

PAIDAR et al.: PLASTIC DEFORMATION OF BICRYSTALS

grain boundary as shown in Fig. 1. When the angle q is zero we get just a continuous PST single crystal without any interface at the grain boundary. For q ⫽ 180°, the grain boundary becomes a true twin interface, and hence there is again no grain boundary in the sense defined above. For the chosen normal to the grain boundary in grain B, the angle q determines both the grain misorientation and inclination of the boundary plane. In agreement with the experimental observations we will assume that the deformation of separated PST crystals is homogeneous on the macroscopic scale. The method applied earlier in our study of the plastic deformation of asymmetrical or asymmetrically loaded cubic bicrystals [9–11] can be modified for the hexagonal crystal. The method is based on the description of anisotropic elastic continuum according to [12] and can be used for the PST bicrystals of arbitrary orientations. The imposed strains of two types can be distinguished in the separated crystals A and B, namely, Ap and eBp the elastic (eijAe and eBe ij ) and plastic (eij ij ) strains. Providing that the difference between the strains in the joint grains is not too large it can be accommodated by the additional compatibility strains (eAij and eBij ) of elastic nature. The conditions of strain compatibility in the coordinate system of the bicrystal, where the axis x2 is perpendicular to the grain boundary ⑀Aij ⫽ ⑀Bij for ij ⫽ 11, 33, 13

(1)

must be fulfilled by the total strains Kp K ⑀Kij ⫽ eKe ij ⫹ eij ⫹ eij for K ⫽ A, B.

The additional stress tensor tij is induced by the difference between the strains in the separated crystals Bp Ae Ap ⌬eij ⫽ eBe ij ⫹ eij ⫺eij ⫺eij .

冢冣

冢

s11⫹ s13⫹ s15⫹

t33 ⫽ ⫺sgn x2 s

⫹ 13

t13

⫹ 15

s

s

⫹ 33

s

⫹ 35

⫹ 35

s

⫹ 55

s

冣冢 冣 ⫺1

⌬e11

⌬e33

(2)

⌬e13

where sij⫹ are sums of elastic compliances in the usual two-index notation in the case of TiAl PST crystals possessing hexagonal symmetry sij⫹ ⫽ sAij ⫹ sBij .

The indexing of the additional stress tij in equation (2) is expressed by the function sgn x2 (x2⬍0 in the crystal A and x2>0 in the crystal B). For more details of the calculations of compatibility stresses see [9]. Notice that the sign for inverse matrix correctly presented in equation (2) is missing in [9] due to a misprint. In order to analyse the effect of the additional compatibility stresses on the activation or suppression of different deformation modes, it is necessary to transform the tensor tij expressed by equation (2) in the coordinate system of the PST bicrystal into the cubic coordinates of all different variants. Since the tetragonality is small, the cubic coordinates can be used but the nature of qualitatively different directions must be respected. In fact, it is sufficient to consider just one cubic coordinate system provided that different types of the crystallographic directions are taken into account with respect to the orientation of the L10 tetragonal lattice. The complete results must be presented for all four types of the {111} planes and for three types of the 具101典 and 具121典 directions. If only the effect of the anisotropy of the elastic deformation is considered, the additional compatibility stress can be evaluated as a fraction of the applied stress resolved to a considered slip or twinning system, that is, as a variation of the Schmid factor that has originally only geometrical character. Besides the elastic constants that have been measured recently for the PST crystal of Ti–49.3at.%Al by Tanaka et al. [13], no other input data are needed for the determination of compatibility stresses according to equation (2). When the difference in the plastic deformation between grains A and B is considered, the extent of the plastic deformation is an additional independent free parameter of the model. Provided that we are interested only in the initiation of new deformation modes due to compatibility stresses at the grain boundary, it is sufficient to take into account only the elastic compatibility stresses as it will be done below.

3. PLASTIC DEFORMATION OF PST SINGLE CRYSTALS

Then the compatibility stresses can be calculated using the expression t11

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Plastic deformation in the g-phase can be carried by four different deformation modes: slip of ordinary dislocations with the 1/2具110] Burgers vectors, slip of the 具101] and 1/2具112] superdislocations and twinning in the 具112] direction [14]. All these deformation modes operate on the {111} close packed planes. The alternating (001) parallel atomic planes in the L10 lattice are composed of the same chemical species (the (001) planes composed of Ti atoms alternate with the parallel planes composed of Al atoms). Hence 1/2具110] are the lattice vectors, whereas 1/2具101] are the Burgers vectors of superpartials as the lattice vectors of L10, identical with the Burgers vectors of superdislocations, are 具101]. Similarly, the 1/2具112] vectors are superlattice vectors, and hence they are

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PAIDAR et al.: PLASTIC DEFORMATION OF BICRYSTALS

the Burgers vectors of superdislocations, and not 1/2具121] which are not the lattice vectors of L10. The observed deformation modes in PST crystals with the A2 ([1¯ 1¯ 2]) orientation for tensile and compressive deformations [5] are marked in bold in Tables 1 and 2. The tables contain the list of slip and twinning systems and respective Schmid factors. The slip and twinning systems are arranged in such a way that the systems in each column have the same Schmid factors, the values for the observed systems are also in bold. Both the slip and twinning systems were systematically generated for all six possible variants. The IT twin variant is equivalent to the IM matrix variant, and similarly, IIT is equivalent to IIIM and IIIT to IIM. Consequently, it is sufficient to present the results only for three matrix variants. The same deformation modes were indeed observed for tensile deformation in the following pairs of variants: IM and IT; IIIM and IIT; IIM and IIIT [5]. Since in principle it should be the case for compression, too, we decided, for simplicity, to include in Table 2 the deformation modes observed in IT under IM (in IIT under IIIM and in IIIT under IIM). Let us emphasize that the magnitude of Schmid factor is decisive for the ordinary dislocations, but in the case of twinning the magnitude of the Schmid factor is important only if the twinning occurs in the twinning sense (which is reversed by the sense of loading). The [101¯ ] superdislocation core can be dissociated essentially in two partials bounding a superlattice intrinsic stacking fault (SISF) with relatively low energy

[101¯ ] ⫽ 1/6[11¯ 2¯ ] ⫹ 1/6[514¯ ]. Similarly, the 1/2[11¯ 2¯ ] superdislocation can be dissociated as 1/2[11¯ 2¯ ] ⫽ 1/6[11¯ 2¯ ] ⫹ 1/3[11¯ 2¯ ]. Due to this asymmetry of the superdislocation cores, they can move more easily in a certain direction than in the reversed one. These properties of deformation modes are manifested in a tension/compression asymmetry as discussed below. The chosen normals to the {111} planes point upwards, that is, they have positive projections to the 具001] direction. Since the [1¯ 1¯ 2] loading direction lies in the domain of the 具001] axis, all four [112¯ ], [11¯ 2¯ ], [1¯ 12¯ ] and [1¯ 1¯ 2¯ ] directions have twinning sense, when the upper crystal is displaced with respect to the lower one. In agreement with the signs of the Schmid factors listed on the last row below the block of the twinning planes and directions in Tables 1 and 2, the [11¯ 2¯ ] (11¯ 1) and [1¯ 12¯ ] (1¯ 11) twinning systems with the largest Schmid factor were observed for tensile straining [5], since they have positive Schmid factors. However, when the sense of loading is reversed for compression, the applied stress acts for these two twinning systems in the antitwinning direction, and the [1¯ 12¯ ] (1¯ 11) and [11¯ 2¯ ] (11¯ 1) twinning systems with the second largest Schmid factors were observed [5] as they are positive. Based on this

Table 1. Observed deformation modes in the A2 (具1¯ 1¯ 2]) PST crystal deformed in tension; the slip and twinning systems are arranged in such a way that the systems in each column have the same Schmid factors marked at the bottom of the upper and lower parts of the table. The directions of ordinary dislocations and of true twinning with the largest Schmid factors are underlined and the observed systems [5] and respective Schmid factors are in bold Tension IM 110 ⴚ11ⴚ2 IIM 101 1⫺2⫺1 IIIM 011 ⫺2⫺11 ⴚ0.272 ⴚ0.314

Slip systems in different variants

ⴚ111 ⫺10⫺1 ⫺1⫺21

0ⴚ11 211

ⴚ1ⴚ10 1ⴚ1ⴚ2

1ⴚ11 10ⴚ1 121

011 ⫺2⫺11

⫺110 ⫺1⫺1⫺2

⫺1⫺11 0⫺1⫺1 2⫺11

101 ⫺121

1⫺10 11⫺2

111 01⫺1 ⫺211

⫺101 1⫺21

11ⴚ1 0⫺1⫺1 ⫺21⫺1

ⴚ110 112

⫺10⫺1 ⫺1⫺21

ⴚ111 0⫺11 211

110 ⫺11⫺2

10⫺1 ⫺1⫺2⫺1

⫺11⫺1 ⫺1⫺10 ⫺112

011 21⫺1

⫺101 1⫺21

111 1⫺10 11⫺2

01⫺1 ⫺211

1ⴚ11 ⴚ1ⴚ10 1⫺1⫺2 ⴚ0.136 0.393

10⫺1 121 0.408 ⫺0.079

0⫺1⫺1 ⫺21⫺1 0.272 ⴚ0.314

11ⴚ1 ⴚ110 112 ⴚ0.408 ⫺0.079

101 1⫺2⫺1 0.136 0.393

0⫺11 ⫺2⫺1⫺1 0.000 ⫺0.314

1⫺1⫺1 ⫺10⫺1 12⫺1 ⫺0.272 0.157

110 1⫺12 0.272 0.157

01⫺1 ⫺211 0.000 0.000

111 ⫺101 1⫺21 0.000 0.000

1⫺10 11⫺2 0.000 0.000

Twinning systems in different variants IM ⫺11⫺2 IIM 1⫺2⫺1 IIIM ⫺2⫺11 ⫺0.314

⫺111 ⫺1⫺21

211

1⫺1⫺2

1⫺11 121

⫺2⫺11

⫺1⫺1⫺2

⫺1⫺11 2⫺11

⫺121

11⫺2

111 ⫺211

1⫺21

11⫺1 ⫺21⫺1

112

⫺1⫺21

ⴚ111 211

ⴚ11ⴚ2

⫺1⫺2⫺1

⫺11⫺1 ⫺112

21⫺1

1⫺21

111 11⫺2

⫺211

1ⴚ11 1ⴚ1ⴚ2 0.393

121 ⫺0.079

⫺21⫺1 ⫺0.314

11⫺1 112 ⫺0.079

1⫺2⫺1 0.393

⫺2⫺1⫺1 ⫺0.314

1⫺1⫺1 12⫺1 0.157

1⫺12 0.157

⫺211 0.000

111 1⫺21 0.000

11⫺2 0.000

PAIDAR et al.: PLASTIC DEFORMATION OF BICRYSTALS

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Table 2. Observed deformation modes in the A2 (具1¯ 1¯ 2]) PST crystal deformed in compression; the slip and twinning systems are arranged in such a way that the systems in each column have the same Schmid factors marked at the bottom of the upper and lower parts of the table. The directions of ordinary dislocations and of true twinning with the largest Schmid factors are underlined and the observed systems [5] and respective Schmid factors are in bold Compression IM 110 ⫺11⫺2 IIM 101 1⫺2⫺1 IIIM 011 ⫺2⫺11 0.272 0.314

Slip systems in different variants

ⴚ111 ⫺10⫺1 ⫺1⫺21

0⫺11 211

ⴚ1ⴚ10 1⫺1⫺2

1ⴚ11 10⫺1 121

011 ⫺2⫺11

⫺110 ⫺1⫺1⫺2

⫺1⫺11 0⫺1⫺1 2⫺11

101 ⫺121

1⫺10 11⫺2

111 01⫺1 ⫺211

⫺101 1⫺21

11ⴚ1 0⫺1⫺1 ⫺21⫺1

ⴚ110 112

⫺10⫺1 ⫺1⫺21

ⴚ111 0ⴚ11 211

110 ⴚ11ⴚ2

10⫺1 ⫺1⫺2⫺1

⫺11⫺1 ⫺1⫺10 ⫺112

011 21⫺1

⫺101 1⫺21

111 1⫺10 11⫺2

01⫺1 ⫺211

1ⴚ11 ⴚ1ⴚ10 1ⴚ1ⴚ2 0.136 ⴚ0.393

10ⴚ1 121 ⴚ0.408 0.079

0⫺1⫺1 ⫺21⫺1 ⴚ0.272 0.314

11ⴚ1 ⴚ110 112 0.408 0.079

101 ⫺2⫺1 ⴚ0.136 ⴚ0.393

0⫺11 ⫺2⫺1⫺1 0.000 0.314

1⫺1⫺1 ⫺10⫺1 12⫺1 0.272 ⫺0.157

110 1⫺12 ⫺0.272 ⫺0.157

01⫺1 ⫺211 0.000 0.000

111 ⫺101 1⫺21 0.000 0.000

1⫺10 11⫺2 0.000 0.000

Twinning systems in different variants IM ⴚ11ⴚ2 IIM 1⫺2⫺1 IIIM ⫺2⫺11 0.314

ⴚ111 ⫺1⫺21

211

1ⴚ1ⴚ2

1ⴚ11 121

⫺2⫺11

⫺1⫺1⫺2

⫺1⫺11 2⫺11

⫺121

11⫺2

111 ⫺211

1⫺21

11⫺1 ⫺21⫺1

112

⫺1⫺21

⫺111 211

⫺11⫺2

⫺1⫺2⫺1

⫺11⫺1 ⫺112

21⫺1

1⫺21

111 11⫺2

⫺211

1⫺11 1⫺1⫺2 ⫺0.393

121 0.079

⫺21⫺1 0.314

11⫺1 112 0.079

1⫺2⫺1 ⫺0.393

⫺2⫺1⫺1 0.314

1⫺1⫺1 12⫺1 ⫺0.157

1⫺12 ⫺0.157

⫺211 0.000

111 1⫺21 0.000

11⫺2 0.000

reasoning, the [1¯ 1¯ 2¯ ] (1¯ 1¯ 1) twinning systems could also be activated but was not observed. Let us note that the 具121] {111} twinning systems do not occur, as they would create high energy complex stacking faults, and thus only the 具112] {111} deformation twins corresponding to SISF are observed. The allowed twinning directions with the largest Schmid factors are underlined in both tables disregarding the twinning sense. The ordinary slip with the largest Schmid factor is observed independently of its sign. The Schmid factors for two types of slip directions are listed below the block of the slip planes and directions. The slip directions in the slip systems with the largest Schmid factors are underlined. Notice that, when the ordinary slip with the largest Schmid factor is active, no other deformation mode occurs on the same {111} plane. Instead, the twinning is observed always in combination with ordinary slip even for the Schmid factor as low as 0.136, but not when the Schmid factor is zero as would be the case of the [1¯ 1¯ 2¯ ] (1¯ 1¯ 1) twinning systems combined with the [1¯ 10] {1¯ 1¯ 1} ordinary slip for compressive straining. Let us discuss now the asymmetry of the motion of the 具101] and 具112] superdislocations. Which Shockley partial of the 1/6具112] type in the dissociation of the 1/2具101] dislocation is leading or trailing one may depend on the definition of the Burgers vector. Each 具121] direction is perpendicular to the 具101] superlattice direction above it in the upper parts of the tables. Hence, the 1/2具101] superpartial can be

dissociated into Shockley partials with the remaining two 具121] directions from which only one is of the SISF 具112] type. Let us consider the sign of projection of the 具101] superlattice Burgers vector to the 具112] direction taking into account the sign of the Schmid factor for the respective 具101] superdislocation. Only when this sign is negative for the slip systems with the largest Schmid factors, the 具101] superdislocation slip was systematically observed. Therefore, from three slip systems in the three matrix variants with the largest Schmid factors that one corresponding to ordinary dislocations is always active but from the remaining two superdislocation slips only one is active according to the sense of acting stress determined as described above. Finally, the 1/2具112] superdislocation slip is observed, when the sign of the last vector component multiplied by the sign of the Schmid factor is positive provided that the level of acting stress is sufficiently high, that is the largest Schmid factor for compression and the second largest for tension. The Schmid factors on the second row at the bottom of the upper part of the tables for the 具112] slip are identical with the Schmid factors for twinning. According to this rule, the 1/2[1¯ 1¯ 2¯ ] (1¯ 1¯ 1) slip should be active in addition to 1/2[1¯ 12¯ ] (1¯ 11) and 1/2[11¯ 2¯ ] (11¯ 1) as it has the same magnitude of the Schmid factor and the correct sign. However, similarly to twinning, the 具112] slip is always observed together with the ordinary slip in the perpendicular direction. The Schmid factor of the [1¯ 10] {1¯ 1¯ 1} ordinary slip conjugate to 1/2[1¯ 1¯ 2¯ ]

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PAIDAR et al.: PLASTIC DEFORMATION OF BICRYSTALS

(1¯ 1¯ 1) is zero and hence these dislocations cannot move. Notice that the 1/2具112] superdislocations on the slip systems with the largest Schmid factor (underlined) would move in the wrong direction under tensile loading. In other words, the active twinning excludes the parallel 具112] slip in agreement with the observation that the propensity of twinning decreases with more frequent occurrence of the 1/2具112] superdislocations [15]. In summary, the complicated deformation picture in PST crystals can be characterized in the following way: the 具101] slip systems with the largest Schmid factors are always activated in the case of the 1/2具110] ordinary dislocations but only if the acting stress has the right sense for the 具101] superdislocations. Twinning can evidently occur only in the twinning direction, and it happens for the A2 loading axis on the twinning systems with the largest Schmid factor for tension and on the second largest for compression. Provided that the applied stress acts in the antitwinning direction, the 1/2具112] slip is activated in the systems with the largest and second largest Schmid factors. Contrary to the slip on the systems with the largest Schmid factors which take place in the directions parallel to the (111) plane of the lamellae, the 具112] twinning or slip and the perpendicular conjugate ordinary slip directions are all inclined to (111). Nevertheless, it was shown that they act together resulting in the total shear deformation parallel to the lamellae. This channelling of plastic deformation parallel to the planes with zero Schmid factor is a remarkable phenomenon that may have broader importance for lamellar structures in general. 4. PLASTIC DEFORMATION OF PST BICRYSTALS

Due to compatibility stresses, the forces acting on dislocations can be modified in both senses. In a simple case, an increase of the stress level in one component crystal is associated with a decrease in the other one. The observed slip and twinning systems in grain B of the A2(90°/0°) bicrystal are summarized in Table 3. Two Schmid factors are presented, the first one for the separated crystals in the upper row and the second one for grain B of the bicrystal in the lower row (in italics), the values for the observed systems are in bold. Since no 具112] slip was detected, the Schmid factors are given only for the 具101] slip in the upper part of Table 3 and for the 具112] twinning in the lower part. Our observations were carried out after tensile straining. The additional slip systems, those not activated in the A2 PST single crystal, are presented in italics and bold. Comparing the Schmid factors for the mono- and bi-crystals, it is seen that the magnitudes for twinning are slightly decreased (4%) for the systems with the largest Schmidt factor and more significantly decreased (12%) for the second largest. In fact, there is also a small increase of 2% for the [1¯ 1¯ 2¯ ] (1¯ 1¯ 1) twinning system but none of the second twinning sys-

tems can be activated as the applied stress is acting in the antitwinning direction in all three of them. Therefore, no change with respect to the single crystal was found in the bicrystal as far as twinning is concerned. The largest Schmid factor for the 具101] slip in grain B of the bicrystal is smaller (8%) but it remains still higher than the second largest which is only about 2% higher. Nevertheless, apparently even such a small difference can cause activation of two new slip systems of ordinary dislocations and one of superdislocations. Notice that the projection of the [101] vector in the [112] twinning direction has positive sign but the Schmid factor of the [101] (111¯ ) slip system in the IIM variant is negative, and hence the superdislocations move in the right direction. The applied stress on the [011] (11¯ 1), [1¯ 01¯ ] (1¯ 11), [1¯ 01¯ ] (11¯ 1¯ ) and [011] (1¯ 11¯ ) slip systems in the IIIM, IIM, IIIM and IIM variants, respectively, have not the correct sign and thus these superdislocations do not move under tensile loading. However, the acting forces have the right sense for the [01¯ 1¯ ] (111¯ ), [01¯ 1¯ ] (1¯ 1¯ 1) and [101] (1¯ 1¯ 1) slip systems in the IIIM, IM and IM variants, respectively, and hence these superdislocation slip systems in principle could be activated. Let us discuss now stress changes in the crystal A (rotated 90°), where the observations of active deformation modes have not been conducted. The Schmid factors for the reference single crystal and both grains of the bicrystal are compared in Table 4. The values for twinning are only slightly increased (2%) for the systems with the largest Schmidt factor and more significantly decreased (7%) for the second largest. In fact, there is also an increase of 7% for the [1¯ 1¯ 2¯ ] (1¯ 1¯ 1) twinning system. But none of the three second twinning systems can be activated as the applied stress is acting in the antitwinning direction in all of them. The situation is finally similar in both grains and hence it can be concluded that no change with respect to the single crystal can be expected in grain A as far as twinning is concerned. The largest Schmid factor for the 具101] slip is only slightly smaller (2%) in grain A but the second largest is about 8% higher in comparison to the single crystal. It can be expected that the respective dislocations active in the single crystal remain active in the bicrystal. Since new slip systems are activated in grain B, where the increase of Schmid factor is smaller than in grain A, their activity can be foreseen also in grain A. The conditions of superdislocation activity following from the asymmetry of their core discussed above for grain B hold also for grain A. The activity of ordinary dislocations and at least some of the permitted superdislocations for the second largest Schmid factor can be anticipated. 5. DEPENDENCE OF COMPATIBILITY STRESSES ON BICRYSTAL MISORIENTATION

It is time and material resources consuming to perform experiments on various bicrystals differing in

PAIDAR et al.: PLASTIC DEFORMATION OF BICRYSTALS

1015

Table 3. Observed deformation modes in the grain B of the A2(90°/0°) bicrystal deformed in tension; the slip and twinning systems are arranged in such a way that the systems in each column have the same Schmid factors marked at the bottom of the upper and lower parts of the table. The Schmid factors in the upper row are for the separated crystals and the Schmid factors in the lower row (in italics) are for the grain B of the bicrystal. The observed systems are in bold, and those differing in the bicrystal from the separated crystals are in bold and italics Tension IM 110 ⫺11⫺2 IIM 101 1⫺2⫺1 IIIM 011 ⫺2⫺11 ⫺0.272 ⴚ0.278

Slip systems in different variants

ⴚ111 ⫺10⫺1 ⫺1⫺21

0ⴚ11 211

ⴚ1ⴚ10 1⫺1⫺2

1ⴚ11 10ⴚ1 121

011 ⫺2⫺11

⫺110 ⫺1⫺1⫺2

⫺1⫺11 0⫺1⫺1 2⫺11

101 ⫺121

1⫺10 11⫺2

111 01⫺1 ⫺211

⫺101 1⫺21

11ⴚ1 0⫺1⫺1 ⫺21⫺1

ⴚ110 112

⫺10⫺1 ⫺1⫺21

ⴚ111 0⫺11 211

110 ⫺11⫺2

10⫺1 ⫺1⫺2⫺1

ⴚ11ⴚ1 ⴚ1ⴚ10 ⫺112

011 21⫺1

⫺101 1⫺21

111 1⫺10 11⫺2

01⫺1 ⫺211

1ⴚ11 ⴚ1ⴚ10 1⫺1⫺2 ⫺0.136 ⴚ0.099

10⫺1 121 0.408 0.377

0⫺1⫺1 ⫺21⫺1 0.272 0.278

11ⴚ1 ⴚ110 112 ⫺0.408 ⴚ0.377

101 1⫺2⫺1 0.136 0.099

0⫺11 ⫺2⫺1⫺1 0.000 0.000

1ⴚ1ⴚ1 ⫺10⫺1 12⫺1 ⫺0.272 ⴚ0.278

110 1⫺12 0.272 0.278

01⫺1 ⫺211 0.000 0.000

111 ⫺101 1⫺21 0.000 0.000

1⫺10 11⫺2 0.000 0.000

Twinning systems in different variants IM ⫺11⫺2 IIM 1⫺2⫺1 IIIM ⫺2⫺11 ⫺0.314 ⫺0.275

⫺111 ⫺1⫺21

211

1⫺1⫺2

1⫺11 121

⫺2⫺11

⫺1⫺1⫺2

⫺1⫺11 2⫺11

⫺121

11⫺2

111 ⫺211

1⫺21

11⫺1 ⫺21⫺1

112

⫺1⫺21

ⴚ111 211

ⴚ11ⴚ2

⫺1⫺2⫺1

⫺11⫺1 ⫺112

21⫺1

1⫺21

111 11⫺2

⫺211

1ⴚ11 1ⴚ1ⴚ2 0.393 0.378

121 ⫺0.079 ⫺0.103

⫺21⫺1 ⫺0.314 ⫺0.275

11⫺1 112 ⫺0.079 ⫺0.103

1⫺2⫺1 0.393 0.378

⫺2⫺1⫺1 ⫺0.314 ⫺0.321

1⫺1⫺1 12⫺1 0.157 0.161

1⫺12 0.157 0.161

⫺211 0.000 0.000

111 1⫺21 0.000 0.000

11⫺2 0.000 0.000

Table 4. Comparison of the Schmid factors in the grain A and B of the A2(90°/0°) bicrystal with the reference PST single crystal; the Schmid factors of the observed slip and twinning systems in the grain B are in bold and italics Tension

Schmid factors for different slip systems

IM

⫺111 110 ⫺10⫺1 Single crystal ⫺0.272 ⫺0.136 Grain A ⫺0.293 ⫺0.106 Grain B ⴚ0.278 ⴚ0.099

0⫺11

⫺1⫺10

1⫺11 10⫺1

011

⫺110

⫺1⫺11 0⫺1⫺1

101

1⫺10

111 01⫺1

⫺101

0.408

0.272

⫺0.408

0.136

0.000

⫺0.272

0.272

0.000

0.000

0.000

0.399

0.293

⫺0.399

0.106

0.000

⫺0.293

0.293

0.000

0.000

0.000

0.377

0.278

ⴚ0.377

0.099

0.000

ⴚ0.278

0.278

0.000

0.000

0.000

Schmid factors for different twinning systems IM

⫺111 ⫺11⫺2 ⫺1⫺21 Single crystal ⫺0.314 0.393 Grain A ⫺0.292 0.400 Grain B ⫺0.275 0.378

211

1⫺1⫺2

1⫺11 121

⫺2⫺11

⫺1⫺1⫺2

⫺1⫺11 2⫺11

⫺121

11⫺2

111 ⫺211

1⫺21

⫺0.079

⫺0.314

⫺0.079

0.393

⫺0.314

0.157

0.157

0.000

0.000

0.000

⫺0.108

⫺0.292

⫺0.108

0.400

⫺0.338

0.169

0.169

0.000

0.000

0.000

⫺0.103

⫺0.275

⫺0.103

0.378

⫺0.321

0.161

0.161

0.000

0.000

0.000

the geometrical parameters such as crystal misorientation and boundary plane inclination. Analytical calculations based on the model described in Section 2 are helpful to assess the trends of the compatibility stresses as a function of the grain boundary crystallographic parameters. In this section, the compatibility stresses will be presented for the bicrystals depicted in Fig. 1 as a function of the angle q. Both grains A

and B are loaded along the 具121典 direction, the grain boundary is parallel in grain B to the {111} plane of the rotational interfaces. Thanks to the symmetry it is not necessary to present the results of calculations for all 24 slip and twinning systems. Since the twinning is observed only for the systems with the first and second largest Schmid factors, three diagrams out of a possible 12 are sufficient. However, the slip hav-

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PAIDAR et al.: PLASTIC DEFORMATION OF BICRYSTALS

ing the third largest (actually, the smallest non-zero) Schmid factor is also activated and hence four other diagrams are required. The slip and twinning systems in each column of the tables presented in the preceding sections, differing in their crystallographic relationship to the L10 tetragonal lattice, are spatially equivalent (parallel slip planes and directions in the fixed coordinate system of the bicrystal). Hence the notation of one variant (e.g. IM) is sufficient for the presentation of the compatibility stresses that are in our model independent of the variant type. Stress modification due to the compatibility conditions at the grain boundary between differently oriented lamellar colonies is normalized by the applied tensile stress. It can be understood as a change of the originally geometrical Schmid factor induced by the presence of the grain boundary. The values of the single crystal Schmid factors are also listed in Table 4. Only the effect of compatibility stresses of the elastic origin is considered, that is, the anisotropy of plastic deformation is not taken into account. All the curves in Figs 2 and 3 for grain B are symmetric with respect to q ⫽ 0° and 90°. For both slip and twinning systems with the largest Schmid factor, the stress m is lower in grain B of the bicrystal than in the single crystal. It might be expected that the quantity m is larger in grain A of the bicrystal than

in the single crystal. Certain increase is indeed observed in the intervals of misorientation close to 0 and 90°, but there are the intervals where m is lower than in the single crystal even in grain A. The curves presented in Fig. 2a are for the [01¯ 1] (1¯ 11) slip system. In grain A, a mirror symmetric curve and negative sign was obtained for [101¯ ] (11¯ 1). Similarly, the curves presented in Fig. 3a are for the [1¯ 2¯ 1] (1¯ 11) twinning system and a mirror symmetric curve but with the same sign was obtained for [2¯ 1¯ 1] (11¯ 1) in grain A. There are two types of the slip and twinning systems with the second largest Schmid factor: The curve for [110] (1¯ 11) in grain A is mirror symmetric with negative sign to that for [1¯ 1¯ 0] (11¯ 1) in Fig. 2b. A similar relationship exists between the second pair of the slip systems [01¯ 1¯ ] (1¯ 1¯ 1) and [101] (1¯ 1¯ 1) in Fig. 2c. Notice that despite a different shape of the curves for grain A in Figs 2b and c, the magnitudes of Schmid factors for the A2(90°/0°) bicrystal are the same (see Table 4). The curve for the [11¯ 2¯ ] (11¯ 1) twinning system in grain A is mirror symmetric to that for [1¯ 12¯ ] (1¯ 11) shown in Fig. 3b. On the other hand, the curves for [1¯ 1¯ 2¯ ] (1¯ 1¯ 1) in Fig. 3c are mirror symmetric with respect to q ⫽ 0 and 90° in both grains A and B. While the magnitude of m in the bicrystal decreases for [1¯ 12¯ ] (1¯ 11) and [11¯ 2¯ ] (11¯ 1) essentially in both grains (see Fig. 3b), it is larger for

Fig. 2. The effect of compatibility conditions on the shear stress m resolved on the {111} plane acting in the 具101典 direction and normalized by the applied tensile stress for the slip system with the first (a), second (b and c) and third (d) largest Schmid factors in two joint grains A and B. q is the misorientation of two PST crystals.

PAIDAR et al.: PLASTIC DEFORMATION OF BICRYSTALS

1017

Fig. 3. The effect of compatibility conditions on the shear stress m resolved on the {111} plane acting in the 具121典 direction and normalized by the applied tensile stress for the slip and twinning system with the first (a) and second (b and c) largest Schmid factors in two joint grains A and B. q is the misorientation of two PST crystals.

[1¯ 1¯ 2¯ ] (1¯ 1¯ 1) again in both grains (Fig. 3c). The sign for all three twinning systems with the second largest Schmid factor is negative for tensile loading. Finally, the curve of the slip system with the third largest Schmid factor for [1¯ 01¯ ] (1¯ 11) in grain A is mirror symmetric with negative sign to that for [011] (1¯ 1¯ 1) depicted in Fig. 2d. In spite of the fact that the low value of m is further decreased in the bicrystal, the activity of respective slip systems of ordinary dislocations was observed not only in the single crystal but also in the A2(90°/0°) bicrystal. 6. DISCUSSION

It has been shown above that the selection of active deformation modes in TiAl alloys with the structure of PST crystals is not governed by the magnitude of the Schmid factor. According to the experimental observations the ordinary dislocations are mobile independently of the sense of loading. On the other hand, the activation of the most stressed 具101] superdislocations in one variant but not in the other one can be explained by the asymmetry of the superdislocation core. Due to this asymmetry described in Section 3, a superdislocation can move easily in a certain sense but not in the opposite one. For the same reason, the superdislocations which are active in tension are not observed in compression and vice versa. It is

argued in [16] that the stress to move a superdislocation with the Shockley partial at the head is larger than to move it in the reversed sense. The interaction forces acting on the partial dislocations in the superdislocation core were analysed in [17]. It was shown that the tangential forces favour a core transformation from the planar to a symmetric wedge shape configuration with SISFs on two inclined octahedral planes. A cross-slip process is further enhanced by the applied stress, when the superdislocation is moving with the Shockley partial at the head, when the angle between the glide and cross-slip planes of the trailing partial is acute. Twinning is naturally asymmetric and hence it is not surprising that the twinning systems with the largest Schmid factors were observed only for tensile straining (see Table 1) while the systems with the opposite sign, having the second largest Schmid factors, were active in compression (see Table 2 where the signs of Schmid factors are already reversed). Notice that the deformation twinning occurs on a given octahedral plane in the variant where the 具101] superdislocations are not active, that is, in the variants IIIM and IIM for tension and in IM for compression. It can be anticipated that twinning takes place when the slip on the most stressed slip system in a given variant does not operate. Contrary to the slip directions of the systems with the largest Schmid factors,

1018

PAIDAR et al.: PLASTIC DEFORMATION OF BICRYSTALS

which lie in the (111) interface plane, the twinning directions are inclined to this plane. In order to preserve the direction of shear deformation parallel to the interfaces, relatively easily mobile ordinary dislocations with the Burgers vectors perpendicular to the twinning direction have to contribute to the plastic flow. Nevertheless, it is surprising that it happens even when the Schmid factor is only about 0.1. The reason for this behaviour is apparently associated with the influence of the lamellar interfaces. A large incompatibility would arise, if a slip operates along the interface on one of its sides and the deformation twinning along an inclined direction on the other side. The described combination of twinning with ordinary slip as a consequence of slip transfer from one lamella to a neighbouring one is compatible with the interpretation of active deformation modes based on the macroscopic straining of PST crystals [5, 14]. For the loading axis lying in the (111) interface plane, including the A2 orientation, the resulting homogeneous deformation is of shear character parallel to the lamellae. Therefore, a PST crystal does not change its dimension perpendicular to the lamellae as it was confirmed by careful strain measurements in [18]. A dependence of plastic deformation on the sense of dislocation motion and twin propagation, that is, on the sense of loading, is clearly very important for cyclic straining [19]. Two types of g domains were distinguished on the basis of observations of the dislocation structures after cyclic loading: mainly deformation twins were formed in the T-type domains and a vein-like dislocation structure was observed in the V-type domains. It is out of the scope of this paper to discuss the complex deformation behaviour arising from various combinations of slip and twinning deformation modes that depends on the amplitude of applied stress in the process of cyclic loading. In a simplified structure presented in Fig. 1, each lamella was considered as one rotational variant. It can be, however, composed of several domains corresponding to different variants separated by narrow domain boundaries perpendicular to the lamellar interfaces. Nevertheless, these details of microstructure have no effect on the results discussed in this paper where the lamellar colonies are treated as averaged homogeneous hexagonal crystals. It should be pointed out that the model described in Section 2 does not take into account the spatial stress variation, in other words, it is strictly valid only for the bicrystals with much larger dimensions parallel to the grain boundary than perpendicular to it. Since the source of additional compatibility stresses is the grain boundary, the magnitude of these stresses is high at the boundary and decreases towards the free surfaces at the side of the bicrystal. The influence of the compatibility stresses on plastic deformation has to be examined in those parts of individual narrow lamellae that are situated in the vicinity of grain boundary. Due to the experimental difficulties to

observe activation of various deformation modes in small parts of different variants, it is very demanding to get information on the spatial variation of compatibility stresses. Such experiments have not yet been carried out and thus the calculated stress modification characterizes correctly and to an adequate extent the effect of grain misorientation on the compatibility stresses. 7. CONCLUSIONS

The activation of different slip and twinning deformation modes in the PST mono- and bi-crystals has been explained by the asymmetric nature of twinning and superdislocation slip. An analytical model for the calculation of the additional stresses arising at the grain boundary as a consequence of the compatibility conditions has been presented and applied to the TiAl bicrystals composed of two lamellar colonies with the interfaces parallel to the loading axis. The predictions of the model agree with the observations of deformation activity in the A2(90°/0°) bicrystal deformed in tension and can be thus used to assess the deformation behaviour in the bicrystals that have not been experimentally investigated. Acknowledgements—One of the authors (V.P.) would like to express his gratitude to the Japanese Society for the Promotion of Science for the fellowship that was awarded to him. This research was also supported by the Grant Agency of the Academy of Sciences (A1010817/1998) and the Ministry of Education (ME190, ME264) of the Czech Republic.

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