Measuring Stress Distributions in Ti-6Al-4V Using Synchrotron X-Ray Diffraction

The material selected for this study was commercially available Ti-6Al-4V plate in a nominal mill annealed condition. The microstructure, as depicted in the backscatter electron micrographs shown in Figure 1, consisted of a bimodal size distribution of equiaxed α (hcp) grains delineated by a relatively small volume fraction (6.5 pct) of β (bcc) phase. The peaks in the α grain-size distribution occur at 9.5 μm and 1.1 μm. The composition of each phase and the overall alloy were estimated by standardless energy-dispersive spectroscopy. The composition of the α and β phases (in weight percentages) was Ti-8.1Al-1.3V and Ti-2.6Al-14.5V, respectively, which were obtained by collecting spot scans from each phase individually. The alloy composition, acquired using a full-frame scan from an area of 2400 μm² was Ti-7.3Al-3.2V. The ODF of each phase, as determined by a Rietveld analysis of the synchrotron X-ray diffraction data,[11,12] is shown in Figure 2, over the associated fundamental zones of the orientation space. We employ the Rodrigues parameterization, \( {\mathbf{r}}={\mathbf{n}}{\text{ tan}}\frac{\phi } {2}, \) where n and ϕ are the axial and angular invariants of the associated rotation.[13] The unit cell axes are attached to the orientation frame using the conventions described by Nye.[14] We define a Cartesian coordinate system on the specimen, so that loading-transverse-normal specimen directions (LD-TD-ND) correspond to x-y-z. The slices shown in Figure 2 are taken perpendicular to a vector along the ND || z direction. The maximum ODF values of 6 to 7 multiples of a uniform distribution (MUD) indicate a moderately strong preferred orientation in both phases.

Electron backscatter diffraction (EBSD) scans were performed in a scanning electron microscope (SEM) equipped with a field emission source and an automated EBSD acquisition system. The SEM was operated at an accelerating voltage of 20 kV and a beam current of 2.39 nA. The EBSD scans were acquired with the sample tilted to 70 deg at a working distance of 21 mm. The first scan was acquired at a step size of 1 μm; it provided the grain-size distribution information and verified the X-ray-based ODFs depicted in Figure 2. Orientation and grain-boundary data from a more highly resolved scan (350-nm step size) are shown in Figure 3. The orientation variation is apparent even within grains. An extensive network of low-angle boundaries (2 to 15 deg) accommodates the change in orientation over the larger grains, while high angle boundaries (>15 deg) are present between the finer-scale grains (Figure 3).

The experimental setup, shown schematically in Figure 5, included a mechanical testing frame with precise specimen alignment, a single-axis goniometer for specimen positioning, and a Mar345¹ online image plate detector.[3] The tensile specimen, shown in Figure 4(a), was taken from the Ti-6Al-4V sheet, so that the LD was parallel to the original rolling direction, and the TD and ND corresponded to the original TD and ND, respectively. Uniaxial deformation was applied along the specimen x axis; hence, the only nonzero macrostress component in the specimen frame is S _xx  = P/A, where P is the applied axial load and A is the cross-sectional area. The macroscopic stress-strain response of the Ti-6Al-4V is shown in Figure 4(b).

The in-situ diffraction experiment involves collecting diffraction data for several specimen orientations at a series of prescribed (constant) macroscopic stress levels. A square 0.5-mm beam of 50 keV X-rays was used to illuminate a volume of grains near the center of the tensile specimen in transmission (Laue) geometry. Additionally, a paste of CeO₂ powder² was applied to the specimen’s downstream face, to provide a fiducial point in each image. The statistical relevance of the diffraction volume was verified by the texture analysis, as discussed in References 8 and 9. The SPFs were collected for applied loads of 0 and 540 MPa, using five images each, measured for ω ∈ [−30, −15, 0, 15, and 30 deg]. The detector was positioned to capture Bragg reflections³ down to ∼1 Å, which encompassed 10 from CeO₂ (fcc), 11 from α-Ti (hcp), and 4 from β-Ti (bcc).

Area detectors facilitate SPF measurements by collecting entire Debye rings for many {hkl} simultaneously.[3,8,9] Figure 6 depicts a typical diffraction image from the Ti-6Al-4V experiments. It is natural to use a polar coordinate system, (ρ′, η′), centered on the transmitted beam, to describe pixel coordinates (Figure 5). Lattice strains are manifested on the detector as radial shifts of the Bragg peaks, in response to the applied stress. In the presence of deviatoric strains, these shifts will have an azimuthal dependence.

The radial detector coordinate is related to the Bragg angle via the instrument geometry, which is discussed further in Section III. For a fixed X-ray wavelength, λ, the Bragg angle is related to the spacing of an associated set of crystallographic planes through Bragg’s law: where \( \bar d_{\mathbf{c}} \) is the mean spacing for the crystallographic family, c ⊥ {hkl}, in the subset of grains satisfying a Bragg condition. If we write the sample relative-scattering vector associated with a diffracted beam as s, the Bragg condition can be written where R ^* represents a one-parameter family of orientations known as a fiber.[15] The shorthand notation, c || s is often used to represent Eq. [2] in the literature. The normal (lattice) strain along c associated with the measured s may then be defined as where \( d_{\mathbf{c}}^0 \) is the reference plane spacing, typically calculated from annealed-state lattice parameters. Equation [3] defines the SPF. In the present context, c is the label, and s is the domain (i.e., the unit sphere). Note that this definition implies antipodal symmetry in \( {\tilde\epsilon}_{\mathbf{c}} ({\mathbf{s}}) \).

Individual radial spectra for distinct values of η′ are produced by performing a polar rebinning of the diffraction image, referred to as “caking.”[16] An example radial spectrum using an azimuthal bin size of Δη′ = 15 deg is shown in Figure 6 (right). The radial spectra for each η′ bin are subsequently converted to angular spectra, using the instrument geometry to convert ρ′ → 2θ (Section III). Lattice-strain data may finally be extracted from the measured 2θ positions of the Bragg peaks in these angular spectra, and assigned to the appropriate SPF.

Powder-diffraction images represent the intersection of the Debye–Scherrer cones with the detector plane, as depicted in Figure 7. As such, the geometry of conic sections may be used to understand the effects of a tilted detector on the detector-relative coordinates, (\(\hat \rho ^{\prime \prime} ,\hat \eta^{\prime \prime} \)), for a strain-free pattern (i.e., the CeO₂).

The instrument geometry for our experimental setup may be effectively modeled by the following four coordinate systems (CSs):

These systems are depicted schematically in Figures 5 and 7. These four coordinate systems are related by the following transformations: where D is the sample to detector distance and the notation R (ϕ n) represents the rotation matrix corresponding to a rotation through angle ϕ about the axis n. The integration CS, \( \left\{ {{\mathbf{\hat X}}^{\prime \prime} {\text{, }}{\mathbf{\hat Y}}^{\prime \prime} {\text{, }}{\mathbf{\hat Z}}^{\prime \prime} } \right\} \), represents the caking coordinate system; its origin represents the initial estimate of the theoretical pattern center. The true pattern center is coincident with the origins of both {X′′, Y′′, Z′′} and {X′, Y′, Z′}. Note that rotations about the detector normal are not considered for several reasons: they only affect the baseline definition of \( \hat \eta^{\prime \prime} \), they are the easiest to minimize in the instrument setup, and they typically cannot be determined using powder-diffraction images.

Image distortion can be attributed to two sources: instrument geometry and intrinsic detector distortions. Table I lists the instrument parameters consistent with our experimental setup and geometric model. It is generally difficult to deconvolve the various causes of image distortion independently. As a result, the only feasible approach is generally via optimization, i.e., the relevant instrument and material parameters are iteratively adjusted (refined), to minimize some objective function based on the errors between the observed and calculated spectra. In this manner, a diffraction image from a strain-free calibrant material, such as powdered Si, LaB₆, or CeO₂, can be used to estimate the following:

Once these parameters are obtained, correction of the diffraction data acquired during the in-situ experiments becomes a matter of transforming the detector coordinates of the raw intensity data, (\( \hat \rho ^{\prime \prime} {\text{, }}\hat \eta ^{\prime \prime} \)) to (ρ′, η′), and converting ρ′ → 2θ. The procedure is performed algorithmically, as follows.

Because the beam, and possibly the detector, can move slightly over the course of the experiment, it is necessary to have a fiducial point in each image. We accomplish this by applying a calibrant powder to the downstream face of the specimen, as described in Section II–B and Eq. [3]. This allows for the subsequent refinement of instrument parameters, as necessary.

The calculation of 2θ _c requires knowledge of the lattice parameters and indices, c ⊥ {hkl}, associated with each peak, along with the wavelength of the incident radiation. When using a layer of calibrant material on the specimen, as described earlier, the offset in scattering centers between the two materials must also be accounted for. This is done with an offset parameter, δ _D , which may be interpreted as the distance along the X-ray beam between the sample and calibrant scattering centers. This can be incorporated into a forward-modeling approach such as Rietveld refinement, which has the benefit of intrinsically accounting for overlapping peaks. This can be particularly important for the study of multiphase samples such as Ti-6Al-4V, in which some degree of overlap between the low-order calibrant and sample reflections is to be expected.

Powder-diffraction spectra may generally be modeled as a superposition of sharp Bragg peaks and a smoothly varying background. Analytic profile functions are used to extract information from these spectra, such as the positions and integrated intensities of the various Bragg peaks. While the procedure of fitting these analytic functions to diffraction spectra may be approached in different ways, its primary objective is to minimize a weighted residual of the form where \( y_i^{{\text{obs}}} \) and \( y_i^{{\text{cal}}} \) are the observed and calculated intensities for the ith angular bin, and w _i is the weight. The weights in Eq. [7] are set to \( w_i = \frac{1} {{y_i^{{\text{obs}}} }} \), as is the case in the standard Rietveld method.[17,18]

For problems in QTA and QSA, sets of spectra, each corresponding to different scattering vectors, are often fit simultaneously. Many analytic functions have been proposed for modeling diffraction spectra. In the present work, a pseudo-Voigt profile function is employed for modeling the Bragg peaks: where A and m are parameters and

While rather simple, this symmetric profile function has been used to successfully model the profiles obtained via synchrotron sources, for a wide variety of cases.[19,20] A simple polynomial is employed to model the background.

The complete calculated intensity profile containing N _c calibrant peaks and N _f “free,” or strained, peaks is then written as where \( p^d = \sum\nolimits_{i = 0}^d {a_i (2\theta )^i } \) is the polynomial background function of degree, d. The third term, \( y^{pV^{\ast}} \), denotes a pseudo-Voigt function having an additional parameter, \( \delta _{2\theta _0 } \), that allows the mean value of the peak, 2θ ₀, to shift by an arbitrary amount associated with the lattice strain for that particular bin. So, 2θ ₀ → 2θ ₀ + \( \delta _{2\theta _0 } \), in Eqs. [9] and [10].

In this analysis, the free parameters describing the shape of each peak, A, m, σ, and Γ, as well as the background coefficients, a _i , are fit to data that have been transformed, using the initial guess for the instrument parameters. If the initial guess is sufficiently close to the optimal solution, then the change in profile shapes and background will be negligible. A program such as the freely available Fit2d (European Synchrotron Radiation Facility, Grenoble, France)[16] can be used to obtain a sufficiently accurate initial guess for the center, tilt, and distance values. With the profile shapes tabulated, optimization for the instrument parameters or strain can be undertaken by minimizing R _y (Eq. [7]) for each image. Table II indicates the parameters that remain free, for each type of image. The first set of images is obtained from a special calibrant specimen, i.e., one geometrically identical to the tensile specimen but containing a window filled with CeO₂ powder. Table III gives the final parameter values for each ω setting of the diffractometer. We also acquire images from the unstrained Ti-6Al-4V sample. The lattice parameters can be determined relatively accurately if the residual strains in the sample are below ∼10⁻⁴, which appears to be a valid assumption for the Ti-6Al-4V. Table IV gives the lattice parameters determined from an unstrained Ti-6Al-4V image. Finally, the data from the strained images are adjusted only for beam movement. The lattice strains for each peak are then extracted from the \( \delta _{2\theta _0 } \) values from the nonlinear optimization. Figure 8 depicts sections from the Ti-6Al-4V + CeO₂ spectrum, from both unstrained and strained images of scattering vectors aligned with the LD and TD. The shift toward smaller values of 2θ (larger lattice spacing) is consistent for {hkl}s aligned with the LD, with larger values of 2θ consistent with scattering vectors aligned with TD. Note that the calibrant powder peaks remain stationary during deformation, as expected.

For most synchrotron sources, E should vary by less than 20 to 30 eV over the course of a typical experiment; hence, it is determined by independent means and subsequently fixed. Furthermore, for diffractometers in high energy with large D configurations suitable for strain measurements, there is a high degree of correlation between small changes in the beam energy, D, and lattice parameters. This precludes the ability to refine these parameters simultaneously. The sequence in which these parameters are determined is given in Table II. Similarly, the detector tilt and radial distortion are not physically expected to change. Appropriate values are obtained by averaging the independently refined values from each unstrained image listed in Table III, which are observed to vary little across the images.

The validity of this method, as well as an estimate of achievable strain resolution, can be verified by examining the deviations between the measured and predicted CeO₂ peak positions. Because we used the “piggyback” calibrant scheme, there are several unstrained CeO₂ peaks in each image. The root-mean-square values of the CeO₂ “pseudostrains” in several postprocessed images were observed to be ∼5 × 10⁻⁵, with maximum magnitudes of ∼1 × 10⁻⁴. There was no discernible ρ or η dependence in the pseudostrain data, which implies the absence of any remaining systematic errors. Because the peak widths for the CeO₂ and Ti (both phases) are on the same order, a conservative estimate of this method’s strain resolution for individual peaks is ∼1 × 10⁻⁴. This is consistent with the reports for similar experimental setups.

The experiments and analysis of the two-phase Ti-6Al-4V data are presented in this article as an example of the variations in the stress state that exist under “simple” uniaxial loading. The resolved shear stress is presented as a simple micromechanical indicator of yielding in soft orientations or fracture in hard orientations. The map in Figure 15 demonstrates the variability in \( \hat \tau _{{\text{rss}}} \) that can be present in a particular microstructure and graphically illustrates which orientations might be at risk under uniaxial loading. Certainly one could extend this analysis by proposing critical values of \( \hat \tau _{{\text{rss}}} \) and possibly determining the volume fractions of the material that falls outside a particular “factor of safety.” Applying the knowledge gained from our experiments to an engineering component, in general, however, must involve a model. The stress distributions and parameter maps we have shown are specific to uniaxial loading. As soon as the macroscopic stress-state changes, the at-risk orientations will change. Interaction with crystal-scale modeling formulations, therefore, is perhaps the greatest utility of these data. As shown in recent work, stress distributions predicted using the QSA technique had excellent agreement with finite element results using polycrystal plasticity[10] for the loading of sintered copper in uniaxial tension. Using simulations, one can test situations that can never be replicated experimentally. In addition to imposing multiaxial macroscopic loading, the model is able to predict local variability that arises due to crystallographic neighborhood and grain-shape effects. One possibility includes using spatially resolved polycrystal plasticity modeling frameworks, such as those based on finite elements or discrete Fourier transform,[28,29] to equilibrate a microstructure, as shown in Figure 15. Orientations in a two- or three-dimensional microstructure could be seeded with stress values from σ( R ), and allowed to come to equilibrium via an elastoviscoplastic constitutive model. This type of synthesis between experiment and simulation could serve to approximate the effects of intergranular stresses, especially at phase boundaries, which could, in turn, illuminate new failure-prone regions. The stress data from the crystalline scale from the σ( R ) experiment, however, represent an important experimental yardstick for measuring the performance of multiscale modeling formulations; favorable comparisons between simulation and experiment build trust in both.