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A method to determine the orientation of the high-temperature beta phase from measured EBSD data for the low-temperature alpha phase in Ti-6Al-4V
Materials Science and Engineering A346 (2003) 50 /59 www.elsevier.com/locate/msea
A method to determine the orientation of the high-temperature beta phase from measured EBSD data for the low-temperature alpha phase in Ti-6Al-4V M.G. Glavicic b,*, P.A. Kobryn a, T.R. Bieler c, S.L. Semiatin a a
Air Force Research Laboratory, Materials and Manufacturing Directorate, AFRL/MLLMP, Wright-Patterson AFB, OH 45433-7817, USA b UES, Inc., 4401 Dayton-Xenia Road, Dayton, OH 45432, USA c Department of Materials Science and Mechanics, Michigan State University, East Lansing, MI 48824-1226, USA Received 7 December 2001; received in revised form 28 May 2002
Abstract A method was developed to determine the orientation of the high-temperature beta phase from measured electron-backscatter diffraction (EBSD) data for the low-temperature alpha phase in Ti-6Al-4V. This technique is an improvement over existing methods because it does not require a priori knowledge of the variant selection process and can accommodate variants from adjacent beta grains being incorporated in the data set submitted for analysis. It is a general method and therefore can be used to examine texture relationships in materials other than Ti-6Al-4V which undergo a burgers-type phase transformation. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Titanium; Phase transformations; Burgers relation; Texture; EBSD
1. Introduction Titanium alloys are commercial materials used in the construction of a wide variety of aircraft components. In pure titanium, a transformation from the body-centered cubic (bcc) beta phase to the hexagonal close-packed (hcp) alpha phase occurs at approximately 883 8C. This a/b transformation is known to follow a well-established burgers relation, viz.: f110gb ==(0001)a
(1)
¯ a h111ib ==h21¯ 10i Based on this orientation relationship and the symmetries of the cubic and hexagonal structures, a total of 12 crystallographically distinct variants of the alpha phase can result from the transformation of a single beta grain. Many commercially available titanium alloys, including the workhorse Ti-6Al-4V, also transform from beta to alpha in this manner. * Corresponding author E-mail address: [email protected] (M.G. Glavicic).
The evolution of crystallographic texture during phase transformations and plastic deformation plays a large role in determining the final microstructure and properties of titanium alloys. Hence, the study of texture evolution has received renewed attention in recent years. In particular, the relationship between the texture of the beta phase and that of the alpha phase is of great interest because many thermomechanical processing steps are conducted within the beta phase field. Therefore, the ability to determine the texture of the parent beta phase is critical. However, such a determination is complicated by the lack of high-temperature X-ray equipment and/or the small volume fraction of retained beta present in most commercial alpha/beta titanium alloys at room temperature. Hence, methods of deducing the texture of the high-temperature beta phase from the room-temperature alpha phase are of great interest. Initial studies of texture in titanium alloys that transform according to a burgers relationship utilized conventional X-ray methods to generate the harmonic series expansion coefficients of the orientation distribution function (ODF) [1 /4]. However, attempts to relate the coefficients from the measured alpha-phase texture
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to those of the parent beta-phase texture were plagued by the need to assume some sort of variant selection process. More recently, an elegant but complex method based on the analysis of electron-backscatter diffraction (EBSD) data was proposed [5 /8]. In this technique, three colonies believed to belong to a single prior-beta grain are judiciously selected. Representative Euler angles for these colonies are determined by EBSD analysis and used as input to a series of equations generated from commutation relationships derived from the burgers relationship and the symmetries of the cubic and hexagonal phases. The orientation of the prior beta grain is then deduced from the solution of these equations. While estimates from the original publication [5] suggested that a solution should be possible for approximately 98% of all possible triplets, solutions were found in practice for only 70 out of 115 triplets
0
This orientation is typically expressed and stored in the form of a set of Euler angles, 81, F , and 82, specified using the Bunge convention. Because EBSD data consist of a set of discrete orientations, they can be used to evaluate the texture of the high-temperature phase from the measured orientation of the low-temperature phase in cases where the phase transformation obeys a specific orientation relationship. The present approach to determine the orientation of a parent beta (bcc) grain from EBSD data for the alpha (hcp) variants within that grain involves transforming the measured alpha-phase orientations back to the possible beta phase orientations using appropriate orientation (rotation) matrices. For a given alpha-phase orientation, the rotation matrix, Pa, is constructed directly from the Euler angles according to the following identity:
sin 8 a1 cos 8 a2 cos8 a1 sin8 a2 cos Fa cos 8 a1 cos 8 a2 sin 8 a2 sin 8 a2 cos Fa a a a a a a @ P cos 8 1 sin 8 2 sin 8 1 cos 8 2 cos F sin 8 a1 sin 8 a2 cos 8 a1 cos 8 a2 cos Fa sin 8 a1 sin Fa cos 8 a1 sin Fa
(/61% efficiency) [7,8]. This poor efficiency was attributed to the fact that the colonies that were selected in the experimental work must not have originated from the same beta grain in all cases. Hence, although this method eliminates the need for an assumption regarding variant selection, its success depends greatly on the proper (manual) selection of colonies or laths. The present work was undertaken to develop a more robust method of determining the parent beta-phase texture from alpha-phase EBSD data that could accommodate the inclusion of data from variants that originated from adjacent prior beta phase grains. This method determines the beta phase orientation by the analysis of possible variants in a reduced Euler space and the application of a Monte-Carlo-type technique to infer those colonies that evolved from the same beta grain. The accuracy and robustness of this technique was demonstrated by analyzing actual EBSD data from a Ti-6Al-4V specimen.
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1 sin 8 a2 sin Fa cos 8 a2 sin Fa A cos Fa
(2)
The steps required to perform the necessary transformation are as follows: 1) Define the crystal orientation for (81, F, 82) / (0, 0, 0) in the hcp structure. 2) Reduce the Euler space populated by the raw EBSD data for each alpha variant by using the symmetry of the hexagonal crystal structure to eliminate redundancy. 3) Determine the minimum number of distinct ways in which an orientation of the hcp phase can be transformed into a single bcc-phase orientation. 4) Determine an approximate solution for the parentphase orientation using a ‘‘minimization-of-misorientation’’ approach. 5) Determine the best choice of the parent-phase orientation using a ‘‘minimization-of-misorientation’’ approach. Each of these steps is described in detail in the following sub-sections.
2. Analysis 2.1. Selection of orientation convention EBSD analysis is a method of characterizing crystallographic texture by determining the crystal orientations at discrete locations within a specimen. The orientation of a crystal at a given location is deduced from the Kikuchi pattern produced by the back-scattered electrons that result from the interaction of a focused electron beam with the inclined surface of the specimen.
To determine potential beta-phase orientations from measured alpha-phase orientations, the burgers relationship must be specified in terms of Euler angles. Before this can be accomplished, the orientation of the hexagonal crystal for the case of (81, F , 82) /(0, 0, 0) must be selected. This is generally done in one of two ways as
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¯ ///RD, [0001]//ND, (b) [21¯ 10] ¯ ///TD, [0001]//ND, (c) orientation of the crystal reference Fig. 1. Possible initial orientations for 81, F , 82 /0: (a) [21¯ 10] frame in relation to the specimen reference frame as a result of the choice of Fig. 1a in conjunction with Eq. (3).
illustrated in Fig. 1. One convention for this choice is to align the right-handed cartesian coordinate system of the crystal, which is defined by the basis vectors (a, b, c) of the crystal’s bravais lattice, with the right-handed cartesian coordinate system describing the specimen reference, transverse, and normal directions (RD /Xs, TD /Ys, ND /Zs) according to the following identities: Zs c Ys c a Xs Ys Zs
(3)
As a result of this choice (Fig. 1a) for the hexagonal ¯ direction is aligned with that of crystal system, the [21¯ 10] the reference direction (RD), and the [0001] is aligned with the normal direction (ND) when (81, F , 82) / (0, 0, 0), as is typically the case in commercial EBSD and texture analysis software. 2.2. Reduction of Euler space For the hexagonal system the space spanned by the Euler angles used to describe an arbitrary orientation measured from EBSD patterns is 05/8a1 B/360, 0 5/ Fa 5/180, and 0 5/8a2 B/360. This space is redundant in that rotations in 8a2 above 608 are a result of the six-fold symmetry of the hexagonal system about the c axis and are not necessary to describe an arbitrary orientation. Since EBSD analysis does not prevent the selection of the redundant orientations, the first step in the conversion process must be to reduce the Euler space populated by the alpha-phase EBSD data and thereby account for the symmetry of the hexagonal phase. This requires that the range of the hexagonal-phase Euler
angles be reduced from 0 5/8a1 B/360, 0 5/Fa 5/180, and 0 5/8a2 B/360 to 0 5/8a1 B/360, 0 5/Fa 5/180, and 05/ 8a2 B/60. The orientationally equivalent set of Euler angles within the reduced angular range is obtained by operating upon the rotation matrix Pa with the six-fold symmetry operators of the hexagonal system, Gai , and calculating the resulting sets of Euler angles. The Gai which places 8a2 into the range 0 5/8a2 B/60 is then used to convert the measured EBSD data into the range 05/ 8a1 B/360, 0 5/Fa 5/180, and 05/8a2 B/60. 2.3. Determination of fundamental transformations In the case of titanium alloys, the relationship describing the orientation of the alpha phase with respect to that of its parent beta grain is given by the burgers relation (Eq. (1)). The burgers relation is ¯ direction and a (0001) plane satisfied whenever a h21¯ 10i in the hexagonal phase coincide with a 111 direction and a {110} plane, respectively, in the cubic phase. A mathematical description of the burgers relationship using rotation matrices was developed in the work of Humbert [6]. An alternative way to view a solution to the problem can be seen if one reviews the mathematical approach developed in this work [6]. For a given orientation of the alpha phase the rotation matrix describing this orientation, Pa, can be transformed to a rotation matrix describing the orientation of the beta phase, Bb, DSbk Bb Saj Pa Sbk
(4) Saj
where and are the set of symmetry operators for the cubic and hexagonal systems shown in Tables 1 and
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Table 1 Rotational symmetry elements of the cubic system 2 3 1 0 0 /E 40 1 05/ 0 0 1 2 3 0 0 1 b 4 5/ /C 1 0 0 31 0 1 0 2 3 0 1 0 b 4 /C 0 15/ 31 0 1 0 0 2 3 1 0 0 b 4 0 15/ /C 4x 0 0 1 0 2 3 0 0 1 b 4 /C 1 0 5/ 4y 0 1 0 0 2 3 0 0 1 b /C 40 1 05/ 2c 1 0 0 b
2 1
3 0 0 1 0 5/ 0 0 1 2 3 0 0 1 b 4 5/ /C 1 0 0 32 0 1 0 2 3 0 1 0 b 4 /C 0 0 15/ 32 1 0 0 2 3 0 0 1 b 4 0 1 05/ /C 4y 1 0 0 2 3 0 1 0 b 4 /C 0 05/ 4z 1 0 0 1 2 3 1 0 0 b 4 0 0 15/ /C 2d 0 1 0
2 3 1 0 0 0 1 05/ 0 0 1 2 3 0 0 1 b 4 5/ /C 1 0 0 33 0 1 0 2 3 0 1 0 b 4 /C 0 0 15/ 33 1 0 0 2 3 0 1 0 b 4 /C 0 05/ 4z 1 0 0 1 2 3 0 1 0 b 4 /C 1 0 0 5/ 2a 0 0 1 2 3 0 0 1 b /C 4 0 1 0 5/ 2e 1 0 0
b 4 /C 2x 0
b /C 4 2y
2 3 1 0 0 0 1 05/ 0 0 1 2 3 0 0 1 b 4 5/ /C 1 0 0 34 0 1 0 2 3 0 1 0 b 4 /C 0 15/ 34 0 1 0 0 2 3 1 0 0 b 4 /C 0 15/ 4x 0 0 1 0 2 3 0 1 0 b 4 /C 1 0 0 5/ 2b 0 0 1 2 3 1 0 0 b /C 4 0 0 15/ 2f 0 1 0 b /C 4 2z
Table 2 Rotational symmetry elements of the hexagonal system 2 1 0 /E equals; 40 1 0 0 2 1 0 a /C 4 0 1 2z 0 0 2 1 0 a 4 /C 21 0 1 0 0 2 1 0 a 40 1 /C 21 0 0 a
3 0 05/ 1 3 0 05/ 1 3 0 0 5/ 1 3 0 0 5/ 1
2 1=2
3 pffiffiffi 3=2 0 1=2 05/ 0 0 1 2 3 pffiffiffi 1=2 3=2 0 pffiffiffi a 4 /C 1=2 05/ 6z 3=2 0 0 1 2 3 pffiffiffi 1=2 3=2 0 pffiffiffi a 4 3=2 1=2 /C 0 5/ 22 0 0 1 2 3 pffiffiffi 1=2 3=2 0 pffiffiffi a 4 /C 0 5/ 22 3=2 1=2 0 0 1 pffiffiffi a 4 3=2 /C 6z
2, respectively, and the matrix D is a rotation matrix which describes the burgers relationship. If the Euler angles used to describe the burgers relation and construct D are selected to be 81 /135, F /90 and 82 / 325, the following commutation relationship between symmetry operators of the cubic and hexagonal phases on the rotation matrix D is applicable. Ca2z D DCb2a
(5)
The impact of this commutation relationship on Eq. (4) can be clearly seen if the hexagonal symmetry operator Ca2z is applied to both sides of Eq. (4). Ca2z DSbk Bb Ca2z Saj Pa
(6)
2 3 pffiffiffi 1=2 3=2 0 1=2 05/ 0 0 1 2 3 pffiffiffi 1=2 3=2 0 pffiffiffi a 4 /C 05/ 3z 3=2 1=2 0 0 1 2 3 pffiffiffi 1=2 3=2 0 p ffiffiffi a 4 3=2 1=2 0 5/ /C 23 0 0 1 2 3 pffiffiffi 1=2 3 =2 0 pffiffiffi a 4 /C 1=2 0 5/ 23 3=2 0 0 1 pffiffiffi a 4 3=2 /C 3z
Eq. (6) can then be rewritten using the commutation relationship defined in Eq. (5). DCb2a Sbk Bb Ca2z Saj Pa
(7)
If Eqs. (4) and (7), respectively, are now rearranged into the formats shown in Eqs. (8) and (9), the application of the symmetry operator Ca2z to the orientation, Saj Pa will change the solution determined by a rotation Cb2a . Sbk Bb D 1 Saj Pa
(8)
Cb2a Sbk Bb D 1 Ca2z Saj Pa
(9)
Because the rotation Cb2a is a cubic symmetry operator, the orientation of the beta grain deduced is unchanged,
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Fig. 2. Graphical representation of inverse transformation solutions: (a) Ta /D 1Ea, (b) Ta /D 1Ca6z , (c) Ta /D 1Ca3z , (d) Ta /D 1Ca21 , (e) Ta /D 1Ca22 , and (f) Ta /D 1Ca23 .
and therefore the solution is symmetrically equivalent. If one then examines the affect of the symmetry operator Ca2z on the set of symmetry operators {Saj } the set of 12 symmetry operators can be split into two sets {Sal } and {Sam }. fSal g (Ea ; Ca6z ; Ca3z ; Ca21 ; Ca22 ; Ca23 )
(10)
fSam g (Ca2z ;
(11)
Ca6z ;
Ca3z ;
Ca21 ;
Ca22 ;
Ca23 )
These two sets are then related to one another through the hexagonal symmetry operator Ca2z in the following way. Sam 1;...;6 Ca2z Sal 1;...;6
(12)
This result leads to the conclusion that for a single alpha phase orientation, there are six unique ways to transform the alpha phase data into the beta phase. The six inverse transformations Tal are given by Tal 1;...;6 D 1 Sal 1;...;6
(13)
Graphically, these six possible Tal solutions are shown in Fig. 2. Examination of the solutions shows that each of the first three solutions (Fig. 2a, b, and c) are distinctly different from one another. Moreover, these solutions (Fig. 2a, b, and c) also have other solutions (Fig. 2e, d, and f, respectively), which are oriented 10.538 from them. Because the solutions a, b and c are distinct from one another, and solutions e, d and f are similar to a, b and c, a two step, iterative approach to solve the problem can be proposed. In the first step, an approximate solution for the orientation of the beta phase is determined using the misorientation of solutions obtained from the variants selected by only
looking at solutions that can be obtained using the inverse transformations described by solutions a, b, and c of Fig. 2. Once an approximate solution for the beta grain is determined using a, b, and c, solutions d, e, and f of Fig. 2 are then considered. If the misorientation amongst the collective data is further reduced by one of these solutions the data are transformed using this inverse transformation instead of the one previously selected in the first step.
2.4. Determination of an approximate solution for the parent-phase orientation To determine an approximate solution to the possible beta-phase orientation, rotation matrices of the same general form shown in Eq. (2), representing three of the six fundamental inverse transformations Tal , are constructed from Eq. (13). These inverse rotation matrices are then used to transform the matrix resulting from the reduction of Euler space (Gai Pa) to the rotation matrices representing the corresponding beta-phase orientations (Bbli ) as follows: Bbli Tal Gai Pa l 1; . . . ; 3 i 1; . . . ; 6
(14)
Three possible sets of Euler angles, corresponding to the solutions of Fig. 2a, b, and c, can then be calculated from the Bbli matrices using the relationships between matrix elements and Euler angles shown in Eq. (2). The required mathematical operations of this method for
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Fig. 3. Graphical representation of Bbli /Tal Gai Pa, Eq. (14): (a) measured alpha orientation, (b) equivalent orientation after symmetry operator Gai has been applied, (c) orientation after inverse transformation Tal /D 1Ea has been applied, (d) orientation after inverse transformation Tal / D 1Ca6z has been applied, (e) orientation after inverse transformation Tal /D 1Ca3z has been applied.
solutions a, b, and c of Fig. 2 and defined by Eq. (8) are shown schematically in Fig. 3. To determine the approximate beta-phase orientation for a given prior beta grain using solutions a, b, and c of Fig. 2, a comparison of the results from different locations within that grain must be made. The transformations which result in the same beta orientation for every location within the prior beta grain are then the correct choices. Thus, the correct choices are those which minimize the misorientation between all points within the grain. The misorientation between two calculated beta-phase orientations can be represented by a misorientation angle, gi , which is calculated directly from the sets of Euler angles representing the two orientations. First, the rotation matrices representing the two orientations, g1 and g2, are constructed from the Euler angles using the relationships in Eq. (2). The misorientation matrix, Dgi , is then constructed from these matrices according to the following equation: 0
Dg11 @ Dgi g2 × g 1 1 Dg21 Dg31
Dg12 Dg22 Dg32
1 Dg13 Dg23 A Dg33
(15)
Finally, the misorientation angle gi is calculated from Dgi as follows: gi cos 1 f(Dg11 Dg22 Dg33 1)=2g
(16)
It is this misorientation angle gi which must be minimized to determine the correct beta orientations for points within a given prior beta grain. For a set of data taken predominantly from a single prior beta grain, the following process is employed to achieve this minimization. First, a Monte Carlo approach is used to compare randomly selected pairs of points and select the transformed orientations which minimize the misorientation angle between the two points using only solutions a, b, and c shown in Fig. 2. For a data set of dimension N , it was determined experimentally that an approximate solution using this approach can be rapidly obtained by performing N2 comparisons. The results from the Monte Carlo calculation are then further improved using a global minimization approach. The sum of the misorientation angles between a given point and every other point in the data set is calculated to determine a total misorientation, gT, for each of the three orientation choices for that point:
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gT Sgi
(17) T
The three values of g are compared, and the orientation choice which results in the lowest total misorientation is selected. This process is repeated for every point in the data set to effectively re-evaluate each set of Euler angles and correctly transform those sets which were transformed incorrectly during the Monte Carlo calculation.
2.5. Determination of the best choice of parent-phase orientation In the final step a global minimization approach identical to that described in the previous section is again employed to obtain the final solution to the problem. In this case only the solutions d, e, and f shown in Fig. 2 are considered. If one of these solutions decreases the total misorientation calculated the data is transformed in this manner; otherwise the data is left untouched. This two-step minimization technique is not sensitive to errors in the EBSD data (such as points from different prior beta grains or unindexed or incorrectly indexed points) provided that the vast majority of the data points is from a single prior beta grain and is correctly indexed. Furthermore, if the number of alpha phase variants contained within the data set being analyzed is limited to three variants from a single prior beta grain, the orientation of the beta phase deduced will be at worst 10.58 from the orientation of the actual beta orientation. Based upon the calculations of a previous work [5], this will occur approximately 2% of the time.
In the case where many prior beta grains are contained within a data set, an iterative approach in which selected areas are parsed from the original data set and analyzed can be used to determine to orientations of the prior individual beta grains. Depending upon the results obtained, the areas selected can then be refined until a single prior beta grain was predominant in the area selected.
3. Validation 3.1. Material and experimental procedure The material used to validate the approach described in Section 2 was 30-mm-thick hot-rolled Ti-6Al-4V plate with a composition (in weight percent) of 6.09 aluminum, 3.95 vanadium, 0.18 iron, 0.19 oxygen, 0.02 carbon, 0.01 nitrogen, and 0.010 hydrogen, balance titanium. The as-received microstructure of the plate was bimodal, i.e. equiaxed alpha in a matrix of transformed beta. A section of this plate was given a heat treatment consisting of 940 8C/20 min/1065 8C/15 min/940 8C/5 min/815 8C/15 min/air cool. A microstructure consisting of alpha colonies within prior-beta grains whose boundaries were decorated by an 8-mmthick layer of alpha (Fig. 4a) was thus produced. The presence of grain-boundary alpha surrounding the complex transformed microstructure made the priorbeta grains easily identifiable. EBSD measurements were collected and analyzed using the method described in Section 2 to establish the efficacy of the overall approach. To this end, the
Fig. 4. Electron backscatter data from SEM: (a) electron backscatter image of area indexed by EBSD, (b) measured orientation map of a-Ti phase.
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heat-treated plate was sectioned and prepared for EBSD analysis using standard metallographic techniques. Alpha-phase EBSD measurements were performed using a Leica Cambridge Stereoscan 360 FE scanning electron microscope affixed with a Noran Voyager EBSD datacollection system and TSL version 2.6 OIM software. The method described in Section 2 was applied to two subsets of the collected EBSD data. The first subset consisted of data points which clearly belonged to several different colonies within a single prior beta grain. The second subset consisted primarily of data points from the same single prior-beta grain as the first subset, but included data points from other prior-beta grains as well. Alpha-phase orientation maps, pole figures, and ODFs were generated and compared to assess the accuracy of the calculation results.
57
3.2. Results One complete prior beta grain is clearly visible in the center of the area that was scanned by the EBSD data collection system (Fig. 4a). The two subsets of data which were input to the beta-phase orientation calculation were chosen to be completely contained within this center grain and to completely contain this center grain, respectively. These subsets are indicated on the alphaphase orientation map of the entire scanned area (Fig. 4b); subset 1 is contained within the small rectangle, and subset 2 is contained within the large rectangle. Several alpha-phase variants were contained within subset 1, as indicated by the vastly different colors present in the alpha-phase orientation map (Fig. 4b). Hence, the ability of the proposed beta-phase orienta-
Fig. 5. Computed b-Ti results from data contained within small rectangle of Fig. 4b: (a) orientation map, (b) pole figures, (c) ODF.
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Fig. 6. Computed b-Ti results from data contained within large rectangle of Fig. 4b: (a) orientation map, (b) pole figures, (c) ODF, (d) variant selection map.
tion calculation could be assessed based on the accuracy with which it was able to transform this subset of data. Upon completion of the beta-phase orientation calculation, the beta-phase orientation map, pole figures, and ODF in Fig. 5 were obtained. While the beta-phase orientation map contained regions of different colors, inspection of the orientations represented by these colors indicated that they were essentially identical. The extremely intense maxima and their location in the pole figures (Fig. 5b) also supported the conclusion that the transformed data represented a single betaphase orientation. Furthermore, the existence of a single maximum in the ODF (Fig. 5c) confirmed the singlecrystal nature of the orientation calculations from subset 1. Hence, the calculation method was validated for the case in which all of the EBSD data comes from a single prior-beta grain.
As shown in Fig. 4b, data from several prior beta grains were indeed contained within subset 2, as indicated by a comparison with Fig. 4a. Hence, the ability of the proposed method to determine the orientation of a given prior beta grain for the case in which a majority, but not all, of the EBSD data is from a single prior-beta grain could be assessed based on the accuracy with which it was able to transform this subset. Upon completion of the beta-phase orientation calculation, the beta-phase map, pole figures, and ODF in Fig. 6 were obtained. A comparison of the beta-phase orientation maps from subsets 1 and 2 (Fig. 5a and Fig. 6a) indicated that the same solution was obtained for the overlapping portion of the two subsets. The orientation representing the prominent prior beta grain, while not the only orientation observed, was also easily discernable in the pole figures and ODF from subset 2
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(Fig. 6b and c). Thus, in all three representations of the transformed data, the orientation of the central grain remained the prominent orientation. Such a result confirms that the orientation of a dominant prior beta grain can be ascertained correctly even for an extreme case in which large amounts of data from several other prior-beta grains are included in the data set. Therefore, this beta-phase orientation calculation method is indeed robust and can be easily implemented without a priori selection of prior-beta grain boundaries or alpha variants within those boundaries.
4. Summary and conclusions A method to determine the crystallographic orientations of prior-beta grains from the orientations of the room-temperature alpha variants within those grains has been developed for titanium alloys. This method requires no a priori knowledge of the variant selection that occurs during the phase transformation or the location of the prior-beta grain boundaries. It is a general approach which can be used to examine phase transformations in other materials as well, provided that a suitable set of transformation matrices can be constructed. This method is substantially more robust than previous methods and allows variants from adjacent prior beta grains to be included in the EBSD data.
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Acknowledgements This work was conducted as part of the in-house research activities of the Metals Processing Group of the Air Force Research Laboratory’s Materials and Manufacturing Directorate. The support and encouragement of the Laboratory management and the Air Force Office of Scientific Research (Dr. C.S. Hartley, program manager) are gratefully acknowledged. Two of the authors, M.G.G. and T.R.B., were supported through Air Force Contracts F33615-99-C-5803 and F33615-94C-5804, respectively.
References [1] M. Humbert, in: H.J. Bunge (Ed.), Directional Properties of Materials, Informationsgesellschaft Verlag, 1988, p. 223. [2] M. Humbert, N. Gey, in: J.A. Szpunar (Ed.), Proc. ICOTOM 12, NRC Research Press, 1999, p. 147. [3] S.V. Divinski, V.N. Dnieprenko, O.M. Ivasishin, Mater. Sci. Eng. A243 (1998) 201. [4] N. Gey, H. Humbert, H. Moustahfid, Scripta Mater. 42 (2000) 525. [5] M. Humbert, H. Moustahfid, F. Wagner, M.J. Philippe, Mater. Sci. Forum 157 (1994) 1225. [6] M. Humbert, F. Wagner, H. Moustahfid, C. Esling, J. Appl. Cryst. 28 (1995) 571. [7] H. Moustahfid, N. Gey, M. Humbert, M.J. Philippe, Metall. Mater. Trans. 28A (1997) 51. [8] H. Moustahfid, M. Humbert, M.J. Philippe, Acta Mater. 45 (1997) 3785.
A method to determine the orientation of the high-temperature beta phase from measured EBSD data for the low-temperature alpha phase in Ti-6Al-4V M.G. Glavicic b,*, P.A. Kobryn a, T.R. Bieler c, S.L. Semiatin a a
Air Force Research Laboratory, Materials and Manufacturing Directorate, AFRL/MLLMP, Wright-Patterson AFB, OH 45433-7817, USA b UES, Inc., 4401 Dayton-Xenia Road, Dayton, OH 45432, USA c Department of Materials Science and Mechanics, Michigan State University, East Lansing, MI 48824-1226, USA Received 7 December 2001; received in revised form 28 May 2002
Abstract A method was developed to determine the orientation of the high-temperature beta phase from measured electron-backscatter diffraction (EBSD) data for the low-temperature alpha phase in Ti-6Al-4V. This technique is an improvement over existing methods because it does not require a priori knowledge of the variant selection process and can accommodate variants from adjacent beta grains being incorporated in the data set submitted for analysis. It is a general method and therefore can be used to examine texture relationships in materials other than Ti-6Al-4V which undergo a burgers-type phase transformation. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Titanium; Phase transformations; Burgers relation; Texture; EBSD
1. Introduction Titanium alloys are commercial materials used in the construction of a wide variety of aircraft components. In pure titanium, a transformation from the body-centered cubic (bcc) beta phase to the hexagonal close-packed (hcp) alpha phase occurs at approximately 883 8C. This a/b transformation is known to follow a well-established burgers relation, viz.: f110gb ==(0001)a
(1)
¯ a h111ib ==h21¯ 10i Based on this orientation relationship and the symmetries of the cubic and hexagonal structures, a total of 12 crystallographically distinct variants of the alpha phase can result from the transformation of a single beta grain. Many commercially available titanium alloys, including the workhorse Ti-6Al-4V, also transform from beta to alpha in this manner. * Corresponding author E-mail address: [email protected] (M.G. Glavicic).
The evolution of crystallographic texture during phase transformations and plastic deformation plays a large role in determining the final microstructure and properties of titanium alloys. Hence, the study of texture evolution has received renewed attention in recent years. In particular, the relationship between the texture of the beta phase and that of the alpha phase is of great interest because many thermomechanical processing steps are conducted within the beta phase field. Therefore, the ability to determine the texture of the parent beta phase is critical. However, such a determination is complicated by the lack of high-temperature X-ray equipment and/or the small volume fraction of retained beta present in most commercial alpha/beta titanium alloys at room temperature. Hence, methods of deducing the texture of the high-temperature beta phase from the room-temperature alpha phase are of great interest. Initial studies of texture in titanium alloys that transform according to a burgers relationship utilized conventional X-ray methods to generate the harmonic series expansion coefficients of the orientation distribution function (ODF) [1 /4]. However, attempts to relate the coefficients from the measured alpha-phase texture
0921-5093/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 2 ) 0 0 5 3 5 - X
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to those of the parent beta-phase texture were plagued by the need to assume some sort of variant selection process. More recently, an elegant but complex method based on the analysis of electron-backscatter diffraction (EBSD) data was proposed [5 /8]. In this technique, three colonies believed to belong to a single prior-beta grain are judiciously selected. Representative Euler angles for these colonies are determined by EBSD analysis and used as input to a series of equations generated from commutation relationships derived from the burgers relationship and the symmetries of the cubic and hexagonal phases. The orientation of the prior beta grain is then deduced from the solution of these equations. While estimates from the original publication [5] suggested that a solution should be possible for approximately 98% of all possible triplets, solutions were found in practice for only 70 out of 115 triplets
0
This orientation is typically expressed and stored in the form of a set of Euler angles, 81, F , and 82, specified using the Bunge convention. Because EBSD data consist of a set of discrete orientations, they can be used to evaluate the texture of the high-temperature phase from the measured orientation of the low-temperature phase in cases where the phase transformation obeys a specific orientation relationship. The present approach to determine the orientation of a parent beta (bcc) grain from EBSD data for the alpha (hcp) variants within that grain involves transforming the measured alpha-phase orientations back to the possible beta phase orientations using appropriate orientation (rotation) matrices. For a given alpha-phase orientation, the rotation matrix, Pa, is constructed directly from the Euler angles according to the following identity:
sin 8 a1 cos 8 a2 cos8 a1 sin8 a2 cos Fa cos 8 a1 cos 8 a2 sin 8 a2 sin 8 a2 cos Fa a a a a a a @ P cos 8 1 sin 8 2 sin 8 1 cos 8 2 cos F sin 8 a1 sin 8 a2 cos 8 a1 cos 8 a2 cos Fa sin 8 a1 sin Fa cos 8 a1 sin Fa
(/61% efficiency) [7,8]. This poor efficiency was attributed to the fact that the colonies that were selected in the experimental work must not have originated from the same beta grain in all cases. Hence, although this method eliminates the need for an assumption regarding variant selection, its success depends greatly on the proper (manual) selection of colonies or laths. The present work was undertaken to develop a more robust method of determining the parent beta-phase texture from alpha-phase EBSD data that could accommodate the inclusion of data from variants that originated from adjacent prior beta phase grains. This method determines the beta phase orientation by the analysis of possible variants in a reduced Euler space and the application of a Monte-Carlo-type technique to infer those colonies that evolved from the same beta grain. The accuracy and robustness of this technique was demonstrated by analyzing actual EBSD data from a Ti-6Al-4V specimen.
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1 sin 8 a2 sin Fa cos 8 a2 sin Fa A cos Fa
(2)
The steps required to perform the necessary transformation are as follows: 1) Define the crystal orientation for (81, F, 82) / (0, 0, 0) in the hcp structure. 2) Reduce the Euler space populated by the raw EBSD data for each alpha variant by using the symmetry of the hexagonal crystal structure to eliminate redundancy. 3) Determine the minimum number of distinct ways in which an orientation of the hcp phase can be transformed into a single bcc-phase orientation. 4) Determine an approximate solution for the parentphase orientation using a ‘‘minimization-of-misorientation’’ approach. 5) Determine the best choice of the parent-phase orientation using a ‘‘minimization-of-misorientation’’ approach. Each of these steps is described in detail in the following sub-sections.
2. Analysis 2.1. Selection of orientation convention EBSD analysis is a method of characterizing crystallographic texture by determining the crystal orientations at discrete locations within a specimen. The orientation of a crystal at a given location is deduced from the Kikuchi pattern produced by the back-scattered electrons that result from the interaction of a focused electron beam with the inclined surface of the specimen.
To determine potential beta-phase orientations from measured alpha-phase orientations, the burgers relationship must be specified in terms of Euler angles. Before this can be accomplished, the orientation of the hexagonal crystal for the case of (81, F , 82) /(0, 0, 0) must be selected. This is generally done in one of two ways as
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¯ ///RD, [0001]//ND, (b) [21¯ 10] ¯ ///TD, [0001]//ND, (c) orientation of the crystal reference Fig. 1. Possible initial orientations for 81, F , 82 /0: (a) [21¯ 10] frame in relation to the specimen reference frame as a result of the choice of Fig. 1a in conjunction with Eq. (3).
illustrated in Fig. 1. One convention for this choice is to align the right-handed cartesian coordinate system of the crystal, which is defined by the basis vectors (a, b, c) of the crystal’s bravais lattice, with the right-handed cartesian coordinate system describing the specimen reference, transverse, and normal directions (RD /Xs, TD /Ys, ND /Zs) according to the following identities: Zs c Ys c a Xs Ys Zs
(3)
As a result of this choice (Fig. 1a) for the hexagonal ¯ direction is aligned with that of crystal system, the [21¯ 10] the reference direction (RD), and the [0001] is aligned with the normal direction (ND) when (81, F , 82) / (0, 0, 0), as is typically the case in commercial EBSD and texture analysis software. 2.2. Reduction of Euler space For the hexagonal system the space spanned by the Euler angles used to describe an arbitrary orientation measured from EBSD patterns is 05/8a1 B/360, 0 5/ Fa 5/180, and 0 5/8a2 B/360. This space is redundant in that rotations in 8a2 above 608 are a result of the six-fold symmetry of the hexagonal system about the c axis and are not necessary to describe an arbitrary orientation. Since EBSD analysis does not prevent the selection of the redundant orientations, the first step in the conversion process must be to reduce the Euler space populated by the alpha-phase EBSD data and thereby account for the symmetry of the hexagonal phase. This requires that the range of the hexagonal-phase Euler
angles be reduced from 0 5/8a1 B/360, 0 5/Fa 5/180, and 0 5/8a2 B/360 to 0 5/8a1 B/360, 0 5/Fa 5/180, and 05/ 8a2 B/60. The orientationally equivalent set of Euler angles within the reduced angular range is obtained by operating upon the rotation matrix Pa with the six-fold symmetry operators of the hexagonal system, Gai , and calculating the resulting sets of Euler angles. The Gai which places 8a2 into the range 0 5/8a2 B/60 is then used to convert the measured EBSD data into the range 05/ 8a1 B/360, 0 5/Fa 5/180, and 05/8a2 B/60. 2.3. Determination of fundamental transformations In the case of titanium alloys, the relationship describing the orientation of the alpha phase with respect to that of its parent beta grain is given by the burgers relation (Eq. (1)). The burgers relation is ¯ direction and a (0001) plane satisfied whenever a h21¯ 10i in the hexagonal phase coincide with a 111 direction and a {110} plane, respectively, in the cubic phase. A mathematical description of the burgers relationship using rotation matrices was developed in the work of Humbert [6]. An alternative way to view a solution to the problem can be seen if one reviews the mathematical approach developed in this work [6]. For a given orientation of the alpha phase the rotation matrix describing this orientation, Pa, can be transformed to a rotation matrix describing the orientation of the beta phase, Bb, DSbk Bb Saj Pa Sbk
(4) Saj
where and are the set of symmetry operators for the cubic and hexagonal systems shown in Tables 1 and
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Table 1 Rotational symmetry elements of the cubic system 2 3 1 0 0 /E 40 1 05/ 0 0 1 2 3 0 0 1 b 4 5/ /C 1 0 0 31 0 1 0 2 3 0 1 0 b 4 /C 0 15/ 31 0 1 0 0 2 3 1 0 0 b 4 0 15/ /C 4x 0 0 1 0 2 3 0 0 1 b 4 /C 1 0 5/ 4y 0 1 0 0 2 3 0 0 1 b /C 40 1 05/ 2c 1 0 0 b
2 1
3 0 0 1 0 5/ 0 0 1 2 3 0 0 1 b 4 5/ /C 1 0 0 32 0 1 0 2 3 0 1 0 b 4 /C 0 0 15/ 32 1 0 0 2 3 0 0 1 b 4 0 1 05/ /C 4y 1 0 0 2 3 0 1 0 b 4 /C 0 05/ 4z 1 0 0 1 2 3 1 0 0 b 4 0 0 15/ /C 2d 0 1 0
2 3 1 0 0 0 1 05/ 0 0 1 2 3 0 0 1 b 4 5/ /C 1 0 0 33 0 1 0 2 3 0 1 0 b 4 /C 0 0 15/ 33 1 0 0 2 3 0 1 0 b 4 /C 0 05/ 4z 1 0 0 1 2 3 0 1 0 b 4 /C 1 0 0 5/ 2a 0 0 1 2 3 0 0 1 b /C 4 0 1 0 5/ 2e 1 0 0
b 4 /C 2x 0
b /C 4 2y
2 3 1 0 0 0 1 05/ 0 0 1 2 3 0 0 1 b 4 5/ /C 1 0 0 34 0 1 0 2 3 0 1 0 b 4 /C 0 15/ 34 0 1 0 0 2 3 1 0 0 b 4 /C 0 15/ 4x 0 0 1 0 2 3 0 1 0 b 4 /C 1 0 0 5/ 2b 0 0 1 2 3 1 0 0 b /C 4 0 0 15/ 2f 0 1 0 b /C 4 2z
Table 2 Rotational symmetry elements of the hexagonal system 2 1 0 /E equals; 40 1 0 0 2 1 0 a /C 4 0 1 2z 0 0 2 1 0 a 4 /C 21 0 1 0 0 2 1 0 a 40 1 /C 21 0 0 a
3 0 05/ 1 3 0 05/ 1 3 0 0 5/ 1 3 0 0 5/ 1
2 1=2
3 pffiffiffi 3=2 0 1=2 05/ 0 0 1 2 3 pffiffiffi 1=2 3=2 0 pffiffiffi a 4 /C 1=2 05/ 6z 3=2 0 0 1 2 3 pffiffiffi 1=2 3=2 0 pffiffiffi a 4 3=2 1=2 /C 0 5/ 22 0 0 1 2 3 pffiffiffi 1=2 3=2 0 pffiffiffi a 4 /C 0 5/ 22 3=2 1=2 0 0 1 pffiffiffi a 4 3=2 /C 6z
2, respectively, and the matrix D is a rotation matrix which describes the burgers relationship. If the Euler angles used to describe the burgers relation and construct D are selected to be 81 /135, F /90 and 82 / 325, the following commutation relationship between symmetry operators of the cubic and hexagonal phases on the rotation matrix D is applicable. Ca2z D DCb2a
(5)
The impact of this commutation relationship on Eq. (4) can be clearly seen if the hexagonal symmetry operator Ca2z is applied to both sides of Eq. (4). Ca2z DSbk Bb Ca2z Saj Pa
(6)
2 3 pffiffiffi 1=2 3=2 0 1=2 05/ 0 0 1 2 3 pffiffiffi 1=2 3=2 0 pffiffiffi a 4 /C 05/ 3z 3=2 1=2 0 0 1 2 3 pffiffiffi 1=2 3=2 0 p ffiffiffi a 4 3=2 1=2 0 5/ /C 23 0 0 1 2 3 pffiffiffi 1=2 3 =2 0 pffiffiffi a 4 /C 1=2 0 5/ 23 3=2 0 0 1 pffiffiffi a 4 3=2 /C 3z
Eq. (6) can then be rewritten using the commutation relationship defined in Eq. (5). DCb2a Sbk Bb Ca2z Saj Pa
(7)
If Eqs. (4) and (7), respectively, are now rearranged into the formats shown in Eqs. (8) and (9), the application of the symmetry operator Ca2z to the orientation, Saj Pa will change the solution determined by a rotation Cb2a . Sbk Bb D 1 Saj Pa
(8)
Cb2a Sbk Bb D 1 Ca2z Saj Pa
(9)
Because the rotation Cb2a is a cubic symmetry operator, the orientation of the beta grain deduced is unchanged,
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Fig. 2. Graphical representation of inverse transformation solutions: (a) Ta /D 1Ea, (b) Ta /D 1Ca6z , (c) Ta /D 1Ca3z , (d) Ta /D 1Ca21 , (e) Ta /D 1Ca22 , and (f) Ta /D 1Ca23 .
and therefore the solution is symmetrically equivalent. If one then examines the affect of the symmetry operator Ca2z on the set of symmetry operators {Saj } the set of 12 symmetry operators can be split into two sets {Sal } and {Sam }. fSal g (Ea ; Ca6z ; Ca3z ; Ca21 ; Ca22 ; Ca23 )
(10)
fSam g (Ca2z ;
(11)
Ca6z ;
Ca3z ;
Ca21 ;
Ca22 ;
Ca23 )
These two sets are then related to one another through the hexagonal symmetry operator Ca2z in the following way. Sam 1;...;6 Ca2z Sal 1;...;6
(12)
This result leads to the conclusion that for a single alpha phase orientation, there are six unique ways to transform the alpha phase data into the beta phase. The six inverse transformations Tal are given by Tal 1;...;6 D 1 Sal 1;...;6
(13)
Graphically, these six possible Tal solutions are shown in Fig. 2. Examination of the solutions shows that each of the first three solutions (Fig. 2a, b, and c) are distinctly different from one another. Moreover, these solutions (Fig. 2a, b, and c) also have other solutions (Fig. 2e, d, and f, respectively), which are oriented 10.538 from them. Because the solutions a, b and c are distinct from one another, and solutions e, d and f are similar to a, b and c, a two step, iterative approach to solve the problem can be proposed. In the first step, an approximate solution for the orientation of the beta phase is determined using the misorientation of solutions obtained from the variants selected by only
looking at solutions that can be obtained using the inverse transformations described by solutions a, b, and c of Fig. 2. Once an approximate solution for the beta grain is determined using a, b, and c, solutions d, e, and f of Fig. 2 are then considered. If the misorientation amongst the collective data is further reduced by one of these solutions the data are transformed using this inverse transformation instead of the one previously selected in the first step.
2.4. Determination of an approximate solution for the parent-phase orientation To determine an approximate solution to the possible beta-phase orientation, rotation matrices of the same general form shown in Eq. (2), representing three of the six fundamental inverse transformations Tal , are constructed from Eq. (13). These inverse rotation matrices are then used to transform the matrix resulting from the reduction of Euler space (Gai Pa) to the rotation matrices representing the corresponding beta-phase orientations (Bbli ) as follows: Bbli Tal Gai Pa l 1; . . . ; 3 i 1; . . . ; 6
(14)
Three possible sets of Euler angles, corresponding to the solutions of Fig. 2a, b, and c, can then be calculated from the Bbli matrices using the relationships between matrix elements and Euler angles shown in Eq. (2). The required mathematical operations of this method for
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Fig. 3. Graphical representation of Bbli /Tal Gai Pa, Eq. (14): (a) measured alpha orientation, (b) equivalent orientation after symmetry operator Gai has been applied, (c) orientation after inverse transformation Tal /D 1Ea has been applied, (d) orientation after inverse transformation Tal / D 1Ca6z has been applied, (e) orientation after inverse transformation Tal /D 1Ca3z has been applied.
solutions a, b, and c of Fig. 2 and defined by Eq. (8) are shown schematically in Fig. 3. To determine the approximate beta-phase orientation for a given prior beta grain using solutions a, b, and c of Fig. 2, a comparison of the results from different locations within that grain must be made. The transformations which result in the same beta orientation for every location within the prior beta grain are then the correct choices. Thus, the correct choices are those which minimize the misorientation between all points within the grain. The misorientation between two calculated beta-phase orientations can be represented by a misorientation angle, gi , which is calculated directly from the sets of Euler angles representing the two orientations. First, the rotation matrices representing the two orientations, g1 and g2, are constructed from the Euler angles using the relationships in Eq. (2). The misorientation matrix, Dgi , is then constructed from these matrices according to the following equation: 0
Dg11 @ Dgi g2 × g 1 1 Dg21 Dg31
Dg12 Dg22 Dg32
1 Dg13 Dg23 A Dg33
(15)
Finally, the misorientation angle gi is calculated from Dgi as follows: gi cos 1 f(Dg11 Dg22 Dg33 1)=2g
(16)
It is this misorientation angle gi which must be minimized to determine the correct beta orientations for points within a given prior beta grain. For a set of data taken predominantly from a single prior beta grain, the following process is employed to achieve this minimization. First, a Monte Carlo approach is used to compare randomly selected pairs of points and select the transformed orientations which minimize the misorientation angle between the two points using only solutions a, b, and c shown in Fig. 2. For a data set of dimension N , it was determined experimentally that an approximate solution using this approach can be rapidly obtained by performing N2 comparisons. The results from the Monte Carlo calculation are then further improved using a global minimization approach. The sum of the misorientation angles between a given point and every other point in the data set is calculated to determine a total misorientation, gT, for each of the three orientation choices for that point:
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gT Sgi
(17) T
The three values of g are compared, and the orientation choice which results in the lowest total misorientation is selected. This process is repeated for every point in the data set to effectively re-evaluate each set of Euler angles and correctly transform those sets which were transformed incorrectly during the Monte Carlo calculation.
2.5. Determination of the best choice of parent-phase orientation In the final step a global minimization approach identical to that described in the previous section is again employed to obtain the final solution to the problem. In this case only the solutions d, e, and f shown in Fig. 2 are considered. If one of these solutions decreases the total misorientation calculated the data is transformed in this manner; otherwise the data is left untouched. This two-step minimization technique is not sensitive to errors in the EBSD data (such as points from different prior beta grains or unindexed or incorrectly indexed points) provided that the vast majority of the data points is from a single prior beta grain and is correctly indexed. Furthermore, if the number of alpha phase variants contained within the data set being analyzed is limited to three variants from a single prior beta grain, the orientation of the beta phase deduced will be at worst 10.58 from the orientation of the actual beta orientation. Based upon the calculations of a previous work [5], this will occur approximately 2% of the time.
In the case where many prior beta grains are contained within a data set, an iterative approach in which selected areas are parsed from the original data set and analyzed can be used to determine to orientations of the prior individual beta grains. Depending upon the results obtained, the areas selected can then be refined until a single prior beta grain was predominant in the area selected.
3. Validation 3.1. Material and experimental procedure The material used to validate the approach described in Section 2 was 30-mm-thick hot-rolled Ti-6Al-4V plate with a composition (in weight percent) of 6.09 aluminum, 3.95 vanadium, 0.18 iron, 0.19 oxygen, 0.02 carbon, 0.01 nitrogen, and 0.010 hydrogen, balance titanium. The as-received microstructure of the plate was bimodal, i.e. equiaxed alpha in a matrix of transformed beta. A section of this plate was given a heat treatment consisting of 940 8C/20 min/1065 8C/15 min/940 8C/5 min/815 8C/15 min/air cool. A microstructure consisting of alpha colonies within prior-beta grains whose boundaries were decorated by an 8-mmthick layer of alpha (Fig. 4a) was thus produced. The presence of grain-boundary alpha surrounding the complex transformed microstructure made the priorbeta grains easily identifiable. EBSD measurements were collected and analyzed using the method described in Section 2 to establish the efficacy of the overall approach. To this end, the
Fig. 4. Electron backscatter data from SEM: (a) electron backscatter image of area indexed by EBSD, (b) measured orientation map of a-Ti phase.
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heat-treated plate was sectioned and prepared for EBSD analysis using standard metallographic techniques. Alpha-phase EBSD measurements were performed using a Leica Cambridge Stereoscan 360 FE scanning electron microscope affixed with a Noran Voyager EBSD datacollection system and TSL version 2.6 OIM software. The method described in Section 2 was applied to two subsets of the collected EBSD data. The first subset consisted of data points which clearly belonged to several different colonies within a single prior beta grain. The second subset consisted primarily of data points from the same single prior-beta grain as the first subset, but included data points from other prior-beta grains as well. Alpha-phase orientation maps, pole figures, and ODFs were generated and compared to assess the accuracy of the calculation results.
57
3.2. Results One complete prior beta grain is clearly visible in the center of the area that was scanned by the EBSD data collection system (Fig. 4a). The two subsets of data which were input to the beta-phase orientation calculation were chosen to be completely contained within this center grain and to completely contain this center grain, respectively. These subsets are indicated on the alphaphase orientation map of the entire scanned area (Fig. 4b); subset 1 is contained within the small rectangle, and subset 2 is contained within the large rectangle. Several alpha-phase variants were contained within subset 1, as indicated by the vastly different colors present in the alpha-phase orientation map (Fig. 4b). Hence, the ability of the proposed beta-phase orienta-
Fig. 5. Computed b-Ti results from data contained within small rectangle of Fig. 4b: (a) orientation map, (b) pole figures, (c) ODF.
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Fig. 6. Computed b-Ti results from data contained within large rectangle of Fig. 4b: (a) orientation map, (b) pole figures, (c) ODF, (d) variant selection map.
tion calculation could be assessed based on the accuracy with which it was able to transform this subset of data. Upon completion of the beta-phase orientation calculation, the beta-phase orientation map, pole figures, and ODF in Fig. 5 were obtained. While the beta-phase orientation map contained regions of different colors, inspection of the orientations represented by these colors indicated that they were essentially identical. The extremely intense maxima and their location in the pole figures (Fig. 5b) also supported the conclusion that the transformed data represented a single betaphase orientation. Furthermore, the existence of a single maximum in the ODF (Fig. 5c) confirmed the singlecrystal nature of the orientation calculations from subset 1. Hence, the calculation method was validated for the case in which all of the EBSD data comes from a single prior-beta grain.
As shown in Fig. 4b, data from several prior beta grains were indeed contained within subset 2, as indicated by a comparison with Fig. 4a. Hence, the ability of the proposed method to determine the orientation of a given prior beta grain for the case in which a majority, but not all, of the EBSD data is from a single prior-beta grain could be assessed based on the accuracy with which it was able to transform this subset. Upon completion of the beta-phase orientation calculation, the beta-phase map, pole figures, and ODF in Fig. 6 were obtained. A comparison of the beta-phase orientation maps from subsets 1 and 2 (Fig. 5a and Fig. 6a) indicated that the same solution was obtained for the overlapping portion of the two subsets. The orientation representing the prominent prior beta grain, while not the only orientation observed, was also easily discernable in the pole figures and ODF from subset 2
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(Fig. 6b and c). Thus, in all three representations of the transformed data, the orientation of the central grain remained the prominent orientation. Such a result confirms that the orientation of a dominant prior beta grain can be ascertained correctly even for an extreme case in which large amounts of data from several other prior-beta grains are included in the data set. Therefore, this beta-phase orientation calculation method is indeed robust and can be easily implemented without a priori selection of prior-beta grain boundaries or alpha variants within those boundaries.
4. Summary and conclusions A method to determine the crystallographic orientations of prior-beta grains from the orientations of the room-temperature alpha variants within those grains has been developed for titanium alloys. This method requires no a priori knowledge of the variant selection that occurs during the phase transformation or the location of the prior-beta grain boundaries. It is a general approach which can be used to examine phase transformations in other materials as well, provided that a suitable set of transformation matrices can be constructed. This method is substantially more robust than previous methods and allows variants from adjacent prior beta grains to be included in the EBSD data.
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Acknowledgements This work was conducted as part of the in-house research activities of the Metals Processing Group of the Air Force Research Laboratory’s Materials and Manufacturing Directorate. The support and encouragement of the Laboratory management and the Air Force Office of Scientific Research (Dr. C.S. Hartley, program manager) are gratefully acknowledged. Two of the authors, M.G.G. and T.R.B., were supported through Air Force Contracts F33615-99-C-5803 and F33615-94C-5804, respectively.
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