research papers
Diffractionbased determination of singlecrystal elastic constants of polycrystalline titanium alloys
^{a}Heinz MaierLeibnitz Zentrum (MLZ), Technische Universität München, Lichtenbergstrasse 1, 85748 Garching, Germany, ^{b}German Engineering Materials Science Centre, HelmholtzZentrum Geesthacht, MaxPlanck Strasse, 21502 Geesthacht, Germany, ^{c}LudwigMaximiliansUniversität München, Department für Geo und Umweltwissenschaften, Theresienstrasse 41, 80333 München, Germany, and ^{d}Institut Laue–Langevin, 71 Avenue des Martyrs, 38042 Grenoble, France
^{*}Correspondence email: alexander.heldmann@frm2.tum.de
Singlecrystal elastic constants have been derived by lattice strain measurements using neutron diffraction on polycrystalline Ti6Al4V, Ti6Al2Sn4Zr6Mo and Ti3Al8V6Cr4Zr4Mo alloy samples. A variety of model approximations for the graintograin interactions, namely approaches by Voigt, Reuss, Hill, Kroener, de Wit and Matthies, including texture weightings, have been applied and compared. A loadtransfer approach for multiphase alloys was also implemented and the results are compared with singlephase data. For the materials under investigation, the results for multiphase alloys agree well with the results for singlephase materials in the corresponding phases. In this respect, all eight elastic constants in the dualphase Ti6Al2Sn4Zr6Mo alloy have been derived for the first time.
Keywords: elastic constants; elasticity; stiffness; multiphase alloys.
1. Introduction
Singlecrystal elastic constants are essential material parameters in fundamental materials science as well as in engineering. In particular, the elastic properties of any polycrystalline bulk material are based on its singlecrystal elastic constants. A variety of micromechanical models have been established to describe the relations between the elastic properties of a polycrystalline bulk material and the singlecrystal elastic constants, the most common applied approaches being those introduced by Voigt (1928), Reuss (1929), Hill (1952), Kroener (1958), de Wit (1997) and Matthies et al. (2001).
In classical stress analysis using diffraction, the residual stresses of alloys are determined by measuring the lattice strains for different sample orientations. Lattice strains and stresses are related by the diffraction elastic constants (DECs), which depend on the direction in the crystal as described by the lattice planes (hkl) of the corresponding Bragg reflections. DECs in turn are based on the singlecrystal elastic constants. Thus, knowledge of the singlecrystal elastic constants is essential for diffractionbased stress analysis.
The conventional stress analysis can be modified in such a way as to apply a defined external stress on the sample (by tensile load or compression) while the lattice strains are measured under various sample orientations. The singlecrystal elastic constants of any polycrystalline material may then be determined by an inversion of the usual calculations applied in the residual stress analysis and fitting by a χ^{2}minimization technique. The feasibility of such an approach was demonstrated by Hauk & Kockelmann (1979) and later applied by various authors (GnäupelHerold et al., 1998; Kisi & Howard, 1998; Singh et al., 1998; Howard & Kisi, 1999; Matthies et al., 2001; Fréour et al., 2005; Stebner et al., 2013; Lunt et al., 2014; Kim et al., 2016) for alloys and ceramics by diffraction under tensile stress or compression. This technique is of particular relevance for any engineering material which cannot be obtained as a single crystal, especially multiphase or highly twinned materials.
In this contribution, we will first review the background to this technique, with particular focus on the measurement geometry and data evaluation. For the validation of our data collection and evaluation procedure, facecentred cubic (f.c.c.) and bodycentred cubic (b.c.c.) phases in steels and cast iron samples were studied. The results are compared with literature data on singlecrystal elastic constants and bulk properties. Approaches for the modelling of graintograin interactions by Voigt (1928), Reuss (1929), Hill (1952), Kroener (1958), de Wit (1997) and Matthies et al. (2001) are applied in the calculations. Since in earlier work the influence of the texture was not investigated in detail, we systematically examine here the influence of texture weightings compared with a quasiisotropic approach. In addition, a loadpartition model is included in the modelling approaches.
Despite the fact that titanium alloys are important highperformance alloys, few data are available concerning DECs or singlecrystal elastic constants. Howard & Kisi (1999) obtained the singlecrystal elastic constants in the near αalloy Ti6Al4V by diffraction on a polycrystalline specimen. In a similar way, Fréour et al. (2005) determined the elastic constants of the b.c.c. phase in the dualphase alloy Ti17.
As the main part of this study, we present the results for the alloys Ti6Al4V (Ti64, near αalloy), Ti3Al8V6Cr4Zr4Mo (Ti38644, βalloy) and Ti6Al2Sn4Zr6Mo (Ti6246, α and βalloy). The elastic constants of both phases in Ti6246 are compared with the results for the singlephase materials.
Ti64 is a highstrength titanium alloy and is considered as the `workhorse' alloy for aerospace applications. It offers excellent strength and toughness up to 673 K combined with good fabricability. As a `near αalloy' it consists mainly of the hexagonal α phase and low amounts of a cubic β phase.
Ti6246 is a very high strength titanium alloy but with lower toughness and weldability than Ti64, although it can be operated at higher temperatures. It consists of two phases, one hexagonal α phase and one b.c.c. β phase.
In general, βalloys have high corrosion resistance and very high strength but a lower elastic modulus than αalloys.
2. Theory
In the following, a general derivation of the current methods is provided. This includes the singlecrystal elastic constants, their relation to the ) is implemented in the evaluation process. A fitting routine for the elastic constants based on the approach by GnäupelHerold et al. (1998) is applied to different micromechanical models.
and their significance for the DECs. For multiphase alloys, a method of quantifying the load transfer from one phase to another as described by Behnken (2003The well known Hooke's law (1) provides the relation between the secondrank tensors of strains ∊ and stresses σ for any material under elastic strain. The proportional constants are given by the fourthrank tensors of elastic compliances A and elastic constants C.
Diffraction studies under the influence of an applied mechanical load enable investigation of the strains perpendicular to particular (hkl) lattice planes, i.e. strains in different crystallographic directions. As outlined in detail by GnäupelHerold et al. (1998), one needs to consider the relations between three coordinate systems. The measurement frame is defined by the scattering vector Q coinciding with the reciprocallattice vector h for each lattice plane (hkl). The load axis determining the applied stress defines the sample frame, while the singlecrystal elastic compliances are expressed in the crystal frame. As illustrated in Fig. 1, the measurement frame L is defined in such a way that the L_{3} axis is parallel to Q. In the sample frame S_{3} is oriented along the load axis in the case of uniaxial tensile (or compression) experiments. As shown in Fig. 1, the orientation between the scattering vector Q and the load axis (S_{3}) is given by the angles ψ and ϕ. S can be transformed into L via the rotation ω as given in equation (2):
The transformation from the crystal frame into the measurement frame is done with the rotation ξ (Behnken, 2003):
In order to ensure the correct orientations of L_{3} to the crystal frame, the rotations are defined by the Euler angles and (GnäupelHerold et al., 1998; Brakman, 1983).
From equation (1) the lattice strains in the measurement frame for a Bragg reflection (hkl) under orientations φ and ψ are given by
with the stress factor F_{ij} given by
are the general singlecrystal compliances expressed in the measurement frame. Therefore A from equation (1) is transformed from the crystal into the measurement frame by the rotation matrix ξ in the following way:
The function f(g) describes the orientation distribution of the grains (ODF) and is typically related to the sample frame (Behnken, 2003), i.e. in equation (5) the integrals are evaluated over all grains contributing to the diffraction signal.
In equation (4) the strains are measured along Q, and the only observed component of the strain tensor is ∊_{33}.
Equation (7) can be transformed with the knowledge of the rotation symmetry of the average tensor and the measurement direction in the measurement frame, yielding
Equation (9) can be transformed into the sample reference frame, which leads to the general equation of stress analysis (Behnken, 2003)
with the DECs s_{1}(hkl) and
An analytical solution of DECs from the singlecrystal elastic constants can be calculated for different crystal symmetries using a model assumption for the graintograin interaction.
In order to establish relationships between bulk and singlecrystalline properties, Voigt assumed that in all grains homogeneous strains appear as a function of an external stress. In this case the general elastic compliances are calculated from the averaged elastic constants 〈c(g)〉^{−1} (Voigt, 1928). Later, Reuss (1929) assumed homogeneous stress in grains; thus A depends on the elastic compliances. Kroener (1958) developed a model based on Eshelby's assumption of spherical elastic inclusions in an isotropic matrix (Eshelby, 1957), which was extended by de Wit (1997). Thus,
The DECs based on the Voigt (1928) model for any crystal symmetries are given in equations (14) and (15):
In equation (15) the singlecrystal elastic constants are represented in Voigt's notation.
The solutions for Reuss's approximation have been developed by Behnken (2003) for arbitrary crystal symmetries:
On the basis of these equations, one can derive the DECs for different crystal symmetries. In the special case of cubic crystal symmetry the DECs can written as
with A_{0} = A_{1111} − A_{1122} − 2A_{1212} and the crystalorientation parameter Γ:
It has been shown by Hill (1952) that the assumptions made by Voigt (1928) and Reuss (1929) are only valid in specific cases, giving an upper and lower limit for the bulk elastic properties for most materials. Therefore, the arithmetic average of the bulk moduli obtained by both models was proposed for more realistic values. However, due to the tensor nature of the elastic constants, the physical relation A = C^{−1} is not granted in this case. Matthies et al. (2001) showed that a geometric average between Voigt (1928) and Reuss (1929) calculations obeys the physical relation A = C^{−1}.
Kroener (1958) introduced a selfconsistent model to solve the case of spherical inclusions in a matrix for cubic symmetries where the G of the matrix is compared with the G_{0} of the inclusion:
with the parameterization
and DECs
de Wit (1997) modified Kroener's approach, recognizing that, by applying statistical averages, the stress is averaged over all spatial directions, whereas the strain is only measured in the direction normal to the diffraction planes. This leads to a slight modification in the description of the parameters (de Wit, 1997):
In a multiphase alloy subjected to elastic strain, a load transfer towards stiffer phases is expected. Diffraction studies reveal the effective elastic constants of the individual phases due to their interactions with the other phases present. Behnken (2003) derived an approach to quantify the load transfer in multiphase systems with different elastic properties. The mean phase stress consists of a macro stressdependent contribution and an independent part denoted by the index 0.
is the fourthrank tensor of the stress transition factors for phase α, σ^{L} is the applied load to the sample and σ^{I} are the residual stresses. To calculate the transition factors, the following equation is derived for an inclusion in a homogeneous matrix (Behnken, 2003):
In the fourthrank tensor equation (27), are the singlecrystal elastic constants of the phase α (in this case the inclusions), I is the unity tensor, and and are the macroscopic elastic moduli and compliances, respectively, based on the following equation:
where the phase fractions of phases α and β are denoted by p^{α, β}, Young's modulus is denoted by E and λ is a Lame constant. w^{−1} depends on the shape of the inclusions and is often called the Eshelby tensor. An analytical solution has been found by Kneer (1965) for spherical and ellipsoidal inclusions:
In the case of spherical inclusions, w^{−1} is isotropic and depends only on the Poisson ratio ν (Kneer, 1965). The stresses of the matrix can be calculated either via the transition factors of the matrix which are obtained via + = I or via the condition + = , where the index m represents the matrix and i the inclusion, and p is the phase fraction of the inclusion.
One needs to address the minimization problem to obtain the singlecrystal elastic constants from diffraction on polycrystalline materials. We choose the method of least squares (χ^{2} minimization) to optimize the elastic compliances by minimizing the sum of squares between deviations of measured and calculated observables.
The first possibility is to derive the DECs from the measured data and subsequently minimize the differences between them and the computed elastic constants (GnäupelHerold et al., 1998),
In equation (30), S is a twodimensional quantity and the total length must be minimized.
The technique for obtaining the values of S_{1}(hkl)_{meas} and is explained in the following section. S_{1}(hkl, A_{ijkl})_{calc} and are computed with the help of equations (16), (18), (14) or (24), depending on the model used.
The second possibility is to express χ^{2} directly in terms of the measured strains , leading to the following expression for χ^{2}:
In principle, both techniques will lead to similar results, but there are important differences between them. Equation (31) is a more specialized form of equation (32), which leads to a large reduction in the summation terms of χ^{2}, e.g. k ≫ n because the S_{i} are combined values from the different measurement directions and only depend on the scattering planes.
Additionally, in both cases the texture of the sample can be taken into account through weightings of either the DECs or the strains directly, based on the multiples of random distribution (m.r.d.) values from the ODF of the different scattering planes involved.
3. Experimental setup
Diffraction studies were carried out using both neutron and highenergy Xray diffractometers. The large gauge volume in neutron experiments yields higher grain statistics compared with Xray studies. On the other hand, highenergy (shortwavelength) Xray experiments enable the recording of complete Debye–Scherrer rings under a continuous increase of the applied load.
A load frame especially designed for elastic anisotropy and texture measurements was used, allowing the load axis to be oriented with respect to the incident beam by a χ rotation as on an Eulerian cradle (Hoelzel et al., 2013).
Neutron diffraction experiments were performed on the highflux powder diffractometer D20 at the Institut Laue–Langevin in Grenoble, France (Hansen et al., 2008) and the highresolution powder diffractometer SPODI at the Heinz MaierLeibnitz Zentrum (MLZ) in Garching near Munich, Germany (Hoelzel et al., 2012). Both instruments operate at a constant wavelength and offer a broad scattering angle range up to 160° in 2Θ. In this setup the position of the sample frame with respect to the measurement frame is given by χ (tilt axis of the load frame) and Ω (rotation axis of the sample table), as shown in Fig. 2. This setup allows a simultaneous analysis of various (hkl) reflections, while the individual orientations between the scattering vectors Q(hkl) and the load axis need to be considered in the data analysis for each reflection individually. By changing the orientation of the load axis by χ and Ω, different angles ψ and φ can be obtained for each (hkl) reflection. In addition, measurements at different χ angles can be used to account for possible texture effects.
The experiments were done using standard tensile samples of 6 or 8 mm in diameter under uniaxial tension; no shear or bending forces were applied, nor did they occur on the sample during the tensile experiments. The macroscopic strain was measured along the load axis using a strain gauge (clipon extensometer Sandner EXA15). For each sample composition, tensile tests were carried out prior to the neutron diffraction studies to determine Young's modulus and the χ angles were performed.
Diffraction patterns were collected at a minimum of three and up to five stress levels, up to a maximum value of about 70% of the At each load step, measurements at a minimum of five and up to tenTo validate our method for data collection and analysis, ferritic (b.c.c.) and austentic (f.c.c) phases in steels and cast iron samples were studied on the SPODI diffractometer at a monochromator angle of 155° using Ge(551) for a wavelength of 1.5483 Å. The instrument offers a good angular resolution to separate the ferrite and austenite peaks and to observe their shifts under load at high precision. The investigations were carried out on ferritic structural steel S235JR, an austenitic stainless steel of AISI type 304 (X5CrNi 1810), duplex steel (X2CrNiMoN 2253) and an austempered ductile iron (ADI) sample consisting of ferrite, austenite and graphite.
The neutron scattering power of titanium alloys is quite poor. This is particularly the case for the β phase of titanium owing to the negative scattering length of Ti and a significant quantity of alloying elements (with positive scattering lengths). Therefore, the investigations on titanium alloys Ti6Al4V (Ti64, near αalloy) and Ti3Al8V6Cr4Zr4Mo (Ti38644, βalloy) were performed on the D20 highflux instrument at a monochromator angle of 90° using Ge(115) to achieve a wavelength of 1.544 Å. This setup offers a good angular resolution, sufficient for the analysis of strains. Samples of 8 mm diameter were measured under uniaxial tension at 0, 10, 20 and 30 kN.
Xray diffraction experiments were carried out on Ti6Al2Sn4Zr6Mo on the High Energy Materials Science beamline (HEMS) at the synchrotron facility DESY in Hamburg, Germany (Schell et al., 2014). An energy of 98.25 keV (corresponding to a wavelength of 0.12619 Å) was used to investigate tensile samples of 6 mm diameter. Since full Debye–Scherrer rings were obtained, no χ rotation was necessary to cover the different orientations of the load axis, which was kept perpendicular to the incident beam, i.e. Ω = 90°. The data were collected in 10 N steps up to 30 kN.
For texture analysis, neutron polefigure measurements were carried out on the STRESSSPEC instrument at the MLZ (Brokmeier et al., 2011) using a Ge(311) monochromator for a wavelength of 1.68 Å. Bragg reflection intensities were measured from 0 to 360° in 5° steps in φ in seven different orientations in χ.
Preparation for electron backscatter diffraction (EBSD) and energydispersive Xray diffraction (EDX) was performed by mechanical cutting and polishing. After cutting, plane grinding was performed with an MDMezzo surface followed by a single fine grinding with a 9 µm diamond suspension abrasive on an MDLargo surface. The first polishing was performed with an MDChem surface and a 0.04 µm collodial silica abrasive and a final step of ion polishing with a Hitachi IM4000PLUS AZTec and CHANNEL5 software. The EDX analysis was carried out in standardless mode.
ion polisher operated at 5 kV and 200 µA for 60 min with sample oscillation. The sample was cut along the cylinder axis, such that the sample normal is the transverse direction of the cylinder. EBSD maps were measured on a Hitachi SU5000 fieldemission scanning electron microscope equipped with an Oxford Instruments NordlysII EBSD detector and an ULTIM MAX EDX detector. The employed acceleration voltage was 20 kV. EBDS and EDX signals were collected simultaneously with the sample inclined by 70° towards the EBSD detector. Data evaluation was performed with the Oxford Instruments4. Data evaluation
For the data evaluation, a software package was developed enabling the choice of all mentioned models and, optionally, the texture implementation. For dualphase alloys, a possible load transfer from one phase to the other can be quantified.
The hkldependent strains = [d(hkl)  d_{0}]/d_{0} were obtained from the changes in the reflection positions under load in a set of diffraction patterns. An example diffraction pattern for Ti64 measured on D20 is shown in Fig. 3. To fit all free variables, at least three or five different Bragg peaks are needed for the cubic and hexagonal symmetries, respectively. The Bragg reflections were chosen to cover the maximum range of crystal directions to track the dependencies of the compliance tensor components.
For the ferritic phase () of the different iron and steel alloys, reflections {110}, {200}, {211}, {220} and {310} were evaluated, and for the austenitic phase () reflections {111}, {200}, {220}, {311} and {222}. For the hexagonal phase in the titanium alloys, all 16 reflections between {100} and {203} were evaluated, and for the cubic phase the {110}, {200}, {211}, {220} and {310} reflections were used for the evaluations.
In uniaxial tension or compression experiments only the σ_{33} component remains nonzero. In this case, equation (10) is simplified to
The equation shown above can be used to fit the DECs if the existing stresses are known or are given by an externally applied stress. The occurring strains for the lattice planes are derived from the shift of the corresponding Bragg peaks (i.e. relative changes in d spacing), while ψ is determined for every Bragg reflection separately by the Ω and χ orientations of the load axis of the tensile rig with respect to the measurement vector Q. An example fit of the (220) DEC of Ti38644 is given in Fig. 4.
It should be emphasized that this evaluation is not affected by intergranular residual stresses, as long as they do not change during the elastic loading, because the lattice strains are given by shifts of the reflections under load rather than by absolute d values.
The evaluation of the pole figures measured on STRESSSPEC were done on the reflections {110} and {200} for , {111} and {200} for , and {002}, {100}, {101}, {103}, {112} and {201} for the hexagonal phase. The pole figures of the first three planes are shown in Fig. 5.
The load transfer is calculated by a selfconsistent scheme until the change in the transition factors after an iteration is smaller than a given limit. The initial values for the elastic constants are used in this approach, with no consideration of the load transfer. The stress factors for the inclusion are calculated with equation (27) defining the stress factors for the matrix. The parameters and σ^{I} in equation (26) are zero in our case, as the effects of residual stresses cancel out during the evaluation procedure as mentioned earlier:
By means of equation (34) the associated stress for each Bragg reflection can be updated and the adjusted elastic constants are calculated. This iteration is repeated until the change in the stress factors is smaller than 10^{−5}.
5. Results
All obtained data were fitted with all available models except the Voigt (1928) model. The calculations based on the approach by Voigt showed large instabilities concerning the fitting routine, led to multiple results or did not converge. Therefore the results for Voigt (1928) are only shown once, for the structural steel S235JR.
The values for the singlephase alloys agree well with available literature data. The results for the ferritic phase are listed in Table 1. Finkel (2016) and GnäupelHerold et al. (1998) used Hill's approach and obtained values of c_{11} = 232.0 GPa, c_{12} = 125.8 GPa and c_{44} = 115.2 GPa, and c_{11} = 224.9 GPa, c_{12} = 122.2 GPa and c_{44} = 120.7 GPa, respectively. Our results of c_{11} = 230.0 GPa, c_{12} = 121.0 GPa and c_{44} = 120.8 GPa using Hill's approximation show good agreement with the literature data. Table 2 illustrates the results for the austenitic phase calculated for different models compared with the literature data. The data published by Ledbetter (1985) yielding c_{11} = 209.0 GPa, c_{12} = 133.0 GPa and c_{44} = 121.0 GPa were predicted on the basis of Kroener's model. Comparison with our values for Kroener's model of 208.0, 135.7 and 116.3 GPa for the isotropic approximation and 204.8, 138.9 and 124.2 GPa for the texture adaptation shows good agreement in both cases.


The results for the duplex steel and ADI are shown in Tables 3 and 4, respectively. Both consist of austenitic and ferritic phases. However, ADI contains a large amount of graphite of approximately 10 vol.% in the form of nodules with Young's and shear moduli of essentially zero. For c_{11} and c_{12} the deviation is of the same order of magnitude as the uncertainty in the values, but for c_{44} the observed deviation is higher. In tensile experiments the parameter c_{44} is related to the shear stresses/strains and is therefore only indirectly accessible with the different orientations covered during the measurements. This leads to higher uncertainties for c_{44} during the determination in the fitting process. The accuracy for c_{44} may be improved by including torsion experiments, as suggested by Woracek et al. (2012).


The tables also reveal larger discrepancies between the models in the Zener anisotropy defined by A = 2c_{44}/(c_{11}  c12) (Zener, 1948) and the c_{12} / c_{11} ratio. In a systematic study of the singlecrystal elastic constants of different monocrystals of austenitic stainless steels and Fe–Cr–Ni alloys, Ledbetter (1985) found that both ratios remain nearly constant at A = 3.53 and c_{12} / c_{11} = 0.635 (Ledbetter, 1985). Owing to the low variance in the values of A and c_{12} / c_{11} in Ledbetter's study, we believe these ratios can be used to estimate the best model for the material under investigation.
In all models it turns out that the texture does not influence the results above their uncertainties. This is most likely due to the elastic measurements being performed on singlecrystal domains and therefore not directly affected by the texture. The only parameter affected is the average strain measured via diffraction. Matthies et al. (2001) concluded that just a couple of thousand grains are enough to ensure the statistical significance of the average strains. Therefore, the isotropic grain orientation assumption is considered to be adequate for the investigated materials as the texture is assumed not to change significantly in the elastic regime with the applied load. Further, our diffraction data show no changes in the Bragg intensities in the elastic regime.
The elastic properties of both phases in duplex steel are very similar, resulting in a load transfer of about 0.3%. Thus, the load transfer approach reveals practically the same elastic constants. A similar behaviour would be expected for ADI but was not investigated further here as the additional graphite phase could not be taken into account.
The results for the obtained elastic constants in the hexagonal α phase in Ti64 and Ti6246 are shown in Fig. 6, while Fig. 7 illustrates the corresponding values for the β phase in Ti38644 and Ti6246. To the best of our knowledge, the analysis of Ti6246 is the first example of deriving all elastic constants in a dualphase (α + β) titanium alloy. Tables 5, 6 and 7 reveal quite similar results for the different model assumptions. In addition, the values obtained for the α + β alloy Ti6246 agree quite well, especially if the load transfer is taken into account, with the corresponding data for the α phase in Ti64 and the β phase in Ti38644, respectively. Table 6 also includes the results of Howard & Kisi (1999) on the Ti64 alloy determined by the Reuss (1929) approach, as well as values obtained by ultrasonic studies on single crystals of pure Ti (Fisher & Renken, 1964).


‡The anisotropy was fixed for these fits. 
Good agreement with ultrasonic data was found in the hexagonal phases for c_{11} and c_{44}. The largest deviation between our results and earlier work is found for c_{33}, and minor deviations in the c_{12} and c_{13} elastic constants. In hexagonal systems the behaviour in the ab plane is completely isotropic owing to the requirement for the c_{66} = (c_{11}  c_{12}) / 2 elastic constant. As we are able to determine c_{11} and c_{12} accurately, this also gives access to the shearing parameter c_{66}. On the other hand, hexagonal systems show additional anisotropies regarding the c axis which influence the precision of parameters c_{12} and c_{13}.
Macroscopic Young's moduli were measured in a series of tensile tests to 106.6 (2.1) GPa for Ti64, to 88.5 (1.6) GPa for Ti38644 and to 113.6 (2.6) GPa for Ti6246. Tables 5, 6 and 7 indicate that there is very good agreement for Ti64 and Ti38644 between these and the values calculated on the basis of the diffraction studies, while somewhat higher values were obtained for Ti6246.
The phase fractions of the dualphase Ti6246 alloys were evaluated from the obtained diffraction patterns using Rietveld analysis (MAUD and FullProf suite) (Lutterotti et al., 1997; RodriguezCarvajal, 1993). For the alloy, phase fractions of 78 and 22% were found for the α and β phases, respectively. The results of phase distribution versus phase composition obtained by EBSD and EDX are shown for the Ti6246 alloy in Fig. 8. The phase fractions determined by EBSD amount to 78 and 22%, the same as for the Rietveld analysis. The main chemical difference between the two phases resides in the Mo content and the compositions are tabulated in Table 8. While the doping elements Sn, Zr and Al are relatively homogeneously distributed, Mo shows an inhomogeneous distribution with elliptical Mopoor regions in the 5–10 µm range (Fig. 8).

The load partitioning applied to the Ti6246 sample indicates a significant load transfer from the β phase to the α phase. The stress in the β phase is reduced by 11.01% and transferred to the α phase, increasing its stress by 3.10% (note that the smaller increase is mainly due to the ratio of phase fractions of around 1:4). The singlecrystal elastic constants corrected for load transfer are shown in Table 7. They reveal a clear shift from the uncorrected values towards the corresponding values of the singlephase samples. The impact of the load transfer in the β phase for the Reuss (1929), Hill (1952) and Matthies et al. (2001) models yields a change of about 10% in most cases, whereas for the Kroener (1958) and de Wit (1997) models the changes in the elastic constants remain below 5%. c_{11} of the β phase was shifted from 143.8 to 124.3 GPa, c_{12} from 78.9 to 66.3 GPa and c_{44} from 43.1 to 37.8 GPa for the Reuss model. The singlephase values of 124.9, 69.9 and 38.0 GPa for the same model almost match the `loadtransfercorrected' ones. Similar behaviour is also observed for all other models. Owing to the lower increase in stress in the α phase the changes remain rather small, but the results achieved with the loadtransfer model match the singlephase values more consistently. The comparatively small changes in the elastic constants when using the Kroener (1958) and de Wit (1997) models are supposedly due to the assumption of inclusions in a matrix, which is essentially the same assumption as used for the load transfer.
Similar elastic constants for α phases in different titanium alloys can be expected owing to the low quantity of alloying elements, shown already by the comparison of Howard & Kisi (1999) with pure titanium. Significant variations in the elastic constants for β phases in different alloys can be found in the literature (Hounkpati et al., 2016; Fréour et al., 2005). This may result from the fact that βalloys contain a large number of different βstabilizing elements in contrast with the α phase. However, our investigations show a larger impact of the modelling between the two investigated β phases.
6. Conclusions
Diffraction experiments are a feasible and powerful route to determine singlecrystal elastic constants from polycrystalline multiphase alloys due to the phase selectivity, as shown in the case of Ti6246.
The dataevaluation procedure included several approximations to consider graintograin interactions in polycrystals based on the approches by Voigt (1928), Reuss (1929), Hill (1952), Kroener (1958), de Wit (1997) and Matthies et al. (2001). In terms of computational stability, the Reuss (1929), Hill (1952) and Matthies et al. (2001) assumptions were the most stable in our data analysis. Concerning the approaches of Kroener (1958) and de Wit (1997), not all possible combinations of the elastic constants led to a solution of the selfconsistent equation (22). The Voigt equations could only be used for the evaluation with very limiting constraints. Good indicators of which models will suit best are the anisotropy values and the c_{12}/c_{11} ratios obtained for the different models. We systematically investigated the effects of the texture on the modelling and find its influence to be smaller than the differences observed for different graintograin interaction models.
For the first time a full analysis of the load transfer has been included in the evaluation of the elastic constants from diffraction data for the example of Ti6246.
For the dualphase alloy Ti6246, the loadtransfer approach allows a direct comparison of measured `effective' elastic constants with `loadtransfercorrected' elastic constants. In this alloy, a significant load relocation of about 14% was found from the β to the α phase. By including the load transfer, the evaluations of the constants of the β phase in the dualphase alloy were significantly shifted towards those of the singlephase alloys, while the effect of the load transfer on the α phase was significantly smaller. In particular, the loadtransfercorrected elastic constants in the α phase of Ti6246 show good agreement with the corresponding values for the near αalloy in Ti64. In addition, for the β phase the loadtransfercorrected constants are in excellent agreement with the corresponding singlephase result for the pure βalloy.
Acknowledgements
We gratefully acknowledge beam time and experimental support at the Heinz MaierLeibnitz Zentrum, the Institut Laue–Langevin and the Deutsches Elektronen Synchrotron. Special thanks are extended to Mr Armin Kriele for valuable support during sample preparation for the metallurgical investigations. The authors would like to thank Dr Roland Hessert (MTU Aero Engines AG) for providing some of the sample material.
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