Measuring Stress Distributions in Ti-6Al-4V Using Synchrotron X-Ray Diffraction
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Open AccessArticle
- First Online:
DOI: 10.1007/s11661-008-9639-6
Abstract
This article presents a quantitative strain analysis (QSA) study aimed at determining the distribution of stress states within a loaded Ti-6Al-4V specimen. Synchrotron X-rays were used to test a sample that was loaded to a uniaxial stress of 540 MPa in situ in the A2 experimental station at the Cornell High Energy Synchrotron Source (CHESS). Lattice-strain pole figures (SPFs) were measured and used to construct a lattice strain distribution function (LSDF) over the fundamental region of orientation space for each phase. A high-fidelity geometric model of the experiment was used to drastically improve the signal-to-noise ratio in the data. The three-dimensional stress states at every possible orientation of each α (hcp) and β (bcc) crystal within the aggregate were calculated using the LSDF and the single-crystal moduli. The stress components varied by 300 to 500 MPa over the orientation space; it was also found that, in general, the crystal stress states were not uniaxial. The maximum shear stress resolved on the basal and prismatic slip systems of all orientations within the α phase, \( \hat \tau _{{\text{rss}}} , \) was calculated to illustrate the utility of this approach for better identifying “hard” and “soft” orientations within the loaded aggregate. Orientations with low values of \( \hat \tau _{{\text{rss}}} , \) which are potential microcrack initiation sites during dwell fatigue conditions, are considered hard and were subsequently illustrated on an electron backscatter diffraction (EBSD) map.
1 Introduction
Newly developed synchrotron X-ray diffraction techniques are changing the manner in which we test the microstructure and, indeed, the micromechanical state within deforming polycrystalline metallic specimens. The combination of rapid collection times and the ability to observe many scattering vectors simultaneously facilitates the performance of thermomechanical processing and performance-like experiments in situ. Experimental techniques have been developed toward this end, to observe both “bulk” populations[1–3] and individual embedded grains.[4–7] The data from such experiments can provide an unparalleled level of detail regarding the evolution of micromechanical states during deformation processes.
Perhaps the most important product from these experiments is the increased understanding of grain-scale deformation partitioning. The measured distributions of crystal (lattice) strains and the related stresses describe the micromechanical state and can be employed to understand important phenomena such as crack initiation and phase transformation. At the scale of statistically representative volumes, these distributions are expected to display a nontrivial orientation dependence. Furthermore, the crystal strains/stresses may in general be quite different from the macroscopic quantities; as such, maximizing the number of independent strain measurements is generally necessary, to best quantify the micromechanical state. Motivated by quantitative texture analysis (QTA), our group has developed a synchrotron X-ray diffraction method for measuring lattice strain pole figures (SPFs). The challenge of such quantitative strain analysis (QSA) experiments is to maximize the number of SPFs measured and the amount of data on each. By employing in-situ mechanical loading, sets of SPFs are acquired at various macroscopic stress values.[3] The SPF data from a polycrystalline aggregate are inverted to form a lattice strain distribution function (LSDF), which is employed within Hooke’s law to calculate the orientation-dependent stress tensor, σ( R ), for every crystal orientation, R , within the aggregate.[8] Previous QSA studies have demonstrated the strong link between σ( R ) and the crystal orientation distribution function (ODF)[9] and have shown a strong correlation between σ( R ) data and crystal-based finite element results.[10] The latter study demonstrated that, even though the stresses computed in the finite element method (FEM) simulation included local effects such as the influence of a crystallographic neighborhood, the orientation-averaged FEM stress distribution compared very well with the experimental σ( R ) results.
In this article, we describe a QSA study aimed at understanding deformation partitioning in Ti-6Al-4V, an important titanium alloy. The large elastic strains and anisotropic single-crystal properties of this alloy make it an ideal and interesting material for these experiments. In the following sections, we describe the study material and experimental method. We focus here on the details of the diffraction data reduction routine that we employ to derive lattice strains from area detector data. We then present the experimental results and demonstrate the utility of the σ( R ) analysis for understanding micromechanically “at-risk” orientations.
2 Material and experiments
The in-situ mechanical tests and synchrotron X-ray experiments performed here for the measurement of SPFs are identical to those described in Reference 3. The experiments were conducted at the A2 experimental station at the Cornell High Energy Synchrotron Source (CHESS). In this section, we describe the study material and present the basics of the experimental method.
2.1 Material
2.2 In-Situ Synchrotron X-Ray Experiments
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 CeO_{2} powder^{2} 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 reflections^{3} down to ∼1 Å, which encompassed 10 from CeO_{2} (fcc), 11 from α-Ti (hcp), and 4 from β-Ti (bcc).
3 Data reduction
The values of 2θ _{ c } are typically measured by fitting an analytic profile function, e.g., pseudo-Voigt, to the peaks in the 2θ spectra. The ability to match the observed and predicted positions of the Bragg peaks with high fidelity is the central task in QSA. However, uncertainties in the instrument geometry and any distortion intrinsic to the detector produce systematic variations in the radial positions of the Bragg peaks, just as strain does. For this reason, all sources of spatial distortions in the diffraction instrument must be quantified a priori, using a strain-free standard, and subsequently deconvolved from the data. While simplified methods, such as approximating each ring as an ellipse, have been employed to fit strained powder patterns with satisfactory results,[1,2] a more sophisticated, self-consistent method is necessary for generating SPFs for LSDF analysis.[3,8] Our approach is to first correct the raw detector data, using the fiducial CeO_{2} pattern, and then fit the η-dependent 2θ spectra from the diffraction pattern to the analytic profile functions. To accomplish this, an accurate geometric model of the diffraction instrument is necessary, as is a method for “unwarping” the raw images, as necessary.
We describe the key points of our data reduction methodology in this section. We begin with the geometric model we employ for the experiment and the way it is employed to correct our raw diffraction data. The scheme we use to calculate lattice strain and to determine SPFs is then described.
3.1 Instrument Geometry
- (1)
the reference CS, {X, Y, Z};
- (2)
the ideal detector CS, {X′, Y′, Z′};
- (3)
the tilted detector CS, {X′′, Y′′, Z′′}; and
- (4)
the integration or raw CS, \( \left\{ {{\mathbf{\hat X}}^{\prime \prime} {\text{, }}{\mathbf{\hat Y}}^{\prime \prime} ,{\mathbf{\hat {\rm Z}}}^{\prime \prime} } \right\}\).
3.2 Correcting for Image Distortion
- (a)
the X-ray wavelength, λ (or equivalently, energy, E);
- (b)
the coordinates of the pattern center (components of t);
- (c)
the sample-to-detector distance, D;
- (d)
the detector nonorthogonality parameters (γ _{ Y′} and γ _{ X′′}); and
- (e)
the radial component of the intrinsic detector distortion,^{4} \( \delta _{\hat \rho \prime \prime } \).
Instrument Parameters Describing the Monochromatic Diffraction Setup with a Two-Dimensional Area Detector*
Parameter |
Description |
Initial Values and Bounds |
---|---|---|
E (keV) |
X-ray energy (equivalent to wavelength) |
49.956 ± 0.02 |
t (cm) |
in-plane origin displacement |
(0.0, 0.0) ± 0.05 |
D (cm) |
sample-to-detector distance |
66.723 ± 5.0 |
δ _{ D } (cm) |
calibrant offset (downstream) |
1.0 ± 0.5 |
γ _{Y′} (deg) |
arccos (X′·X′′) |
0.0 ± 1.0 |
γ _{X′′} (deg) |
arccos (Y′·Y′′) |
0.0 ± 1.0 |
\( \delta _{\hat \rho \prime \prime } \) (cm) |
isotropic radial offset |
0.0 ± 0.2 |
- (1)
Apply \( \delta _{\hat \rho ^{\prime \prime} } \) to \( \hat \rho ^{\prime \prime} \).
- (2)
Transform (\( \hat \rho ^{\prime \prime} {\text{, }}\hat \eta ^{\prime \prime} \)) to (ρ′′, η′′), using t.
- (3)
Transform (ρ′′, η′′) to d′′ = ⌊x′′, y′′, z′′⌋.
- (4)
Transform d′′ → d′ = ⌊x′, y′, z′⌋, using R (γ _{ X′′} X′′) and R(γ _{ Y′} Y′).
- (5)
Transform d ′ → d = ⌊x, y, z⌋, using D.
- (6)
Calculate 2θ from d as arccos (−Z·d).
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.
3.3 Fitting Corrected Data
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.
Free Parameters during the Fitting of Each Image Type
Image Type |
Free Parameters |
---|---|
Calibrant image |
t, D, γ _{Y′}, γ _{X′′}, \( \delta _{\hat \rho^{\prime \prime} } \) |
Unstrained image |
t, δ _{ D }, a _{ α }, a _{ β }, c _{ α } |
Strained image |
t |
Optimal Solutions for the Suite of Calibrant and Unstrained Images, Following the Schedule Given in Table II*
Parameter |
−30 Deg |
−15 Deg |
0 Deg |
15 Deg |
30 Deg |
---|---|---|---|---|---|
t _{ x } (mm) |
0.004 |
0.003 |
0.003 |
0.002 |
0.003 |
t _{ y } (mm) |
0.003 |
0.004 |
0.005 |
0.005 |
0.005 |
D (cm) |
67.270 |
67.281 |
67.287 |
67.318 |
67.310 |
δ _{ D } (mm) |
1.267 |
1.221 |
1.126 |
1.207 |
1.337 |
γ _{Y′} (deg) |
−0.457 |
−0.439 |
−0.446 |
−0.447 |
−0.439 |
γ _{X′′} (deg) |
0.592 |
0.591 |
0.584 |
0.584 |
0.583 |
\( \delta _{\hat \rho \prime \prime } \) (mm) |
0.083 |
0.084 |
0.081 |
0.084 |
0.081 |
Initial and Refined Lattice Parameters for the Ti-6Al-4V Specimen from the Corrected Reference Image (in Å)*
Phase |
Initial Values and Bounds |
Refined Values |
---|---|---|
α-Ti |
(2.9226, 4.6676) ± 0.02 |
(2.9323, 4.6844) |
β-Ti |
3.2131 ± 0.02 |
3.2247 |
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 CeO_{2} peak positions. Because we used the “piggyback” calibrant scheme, there are several unstrained CeO_{2} peaks in each image. The root-mean-square values of the CeO_{2} “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 CeO_{2} 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.
4 Results
4.1 Distribution of Crystal Stresses, σ( R )
In addition to requiring coincidence between the measured lattice strains, \( {{\tilde \epsilon }}_{\mathbf{c}}^M \), and those recalculated from the LSDF, \( {{\tilde \epsilon }}_{\mathbf{c}}^R \), we also penalize gradients in the LSDF and dilation over the orientation space with the second and third terms, respectively, in Eq. [13]. Guidelines related to the choice of the weighting parameters ξ and κ are discussed in detail in Reference 8. For the Ti-6Al-4V data, we employed values of ξ = 0.05 and κ = 100, for both phases.
The constraints based on the macroscopically applied stress, Eqs. [14] and [15], were first introduced by Bernier et al.[9] In the case of a multiphase system, their application becomes slightly more ambiguous. Because the α phase accounts for 93.5 pct of the volume and percolates completely through the sample, as seen in Figure 1, however, we felt justified in assuming that its calculated macrostress should be constrained to match the applied value closely. Due to its uniform distribution, the β phase was also constrained to match the applied stress.
Both the measured SPFs and the SPFs recalculated from the LSDFs for the α and β phases are depicted in Figures 9 and 10, respectively.
Nonzero Components of the Elastic Stiffness Tensor for Each Phase Employed in the Determination of σ( R )*
Modulus (GPa) |
\( {\mathbb{C}}_{11} \) |
\( {\mathbb{C}}_{22} \) |
\( {\mathbb{C}}_{33} \) |
\( {\mathbb{C}}_{12} \) |
\( {\mathbb{C}}_{13} \) |
\( {\mathbb{C}}_{23} \) |
\( {\mathbb{C}}_{44} \) |
\( {\mathbb{C}}_{55} \) |
\( {\mathbb{C}}_{66} \) |
---|---|---|---|---|---|---|---|---|---|
α |
141 |
141 |
163 |
76.9 |
57.9 |
57.9 |
48.7 |
48.7 |
102.6 |
β |
135 |
135 |
135 |
113 |
113 |
113 |
54.9 |
54.9 |
54.9 |
5 Discussion
The stress distributions depicted in Figures 11 and 12 exhibit some interesting trends. The first indicator that significant variations exist is the 500 MPa range in the σ _{ xx } component (LD) in each phase. The off-axis stresses, σ _{ yy } and σ _{ zz }, also vary by over 500 MPa and are both positive and negative. These results illustrate that the stress state at many orientations is not uniaxial tension. Finally, from the nonzero values of the shear components of stress, we see that the principal directions of the stress-state distributions do not necessarily align with the LD, TD, and ND. Armed with the full lattice-strain and crystal-stress tensors at every point in the orientation space, we can begin to address the crystal-level material response and identify worst-case orientations. Here, we examine these using the α-phase σ(R), to examine potentially hard and soft orientations.
5.1 Resolved Shear Stress
An extreme inelastic anisotropy creates a virtual inelastic inextensibility along the c axis in hcp materials such as α-titanium.[21] The most easily activated slip systems are the basal {0001} \( \left\langle {11\bar 20} \right\rangle \) and prismatic \( \left\{ {10\bar 10} \right\} \) \( \left\langle {11\bar 20} \right\rangle \) families, neither of which accommodate extension of the c axis. There are pyramidal systems available in Ti-6Al-4V that accom.modate c + a slip; however, they are approximately 3 to 5 times harder to activate. Thus, a hard orientation results when relatively small values of shear stress, as compared to the critical value, are resolved on the basal and prism planes. The experiments were conducted on Ti-6Al-4V but hard orientations have been implicated in many fatigue or fracture-related failures in titanium alloys. A particularly deleterious condition known as dwell fatigue occurs in alloys such as Ti-6Al-2Sn-4Zr-2Mo.[22–25] Dwell fatigue results in significantly lower lives for loading situations containing a tensile hold component.[26] We seek to leverage σ( R ), to identify hard orientations in the α phase, which could potentially be used to identify failure-prone grains in an aggregate.
6 Summary and direction
A synchrotron X-ray diffraction method for measuring the orientation-averaged stress tensor in each phase of a loaded two-phase Ti-6Al-4V specimen was presented. The method consists of measuring lattice SPFs from a Ti-6Al-4V specimen deformed in situ within the experimental station. The lattice strains are converted to a lattice strain tensor field over orientation space using a method motivated by QTA, which we refer to as QSA. The stress distribution, σ( R ), was determined using Hooke’s law and the single-crystal elastic moduli for each phase. A precise geometric model of the experiment was constructed, to account for experimental artifacts that conspire to create large errors in the experimental data.
The stress distributions in each phase of the loaded Ti-6Al-4V sample, which were determined using the QSA analysis, exhibited significant variations in both magnitude and direction. Stress states within the aggregate varied significantly from the macroscopically applied stress. In each phase, fluctuations equal to the magnitude of the applied macroscopic stress (540 MPa) were apparent for all normal stress values, and 300 MPa fluctuations were observed in the shear components.
The utility of the σ(R) approach was demonstrated by calculating the resolved shear stress in the α phase on the basal plane, \( \hat \tau _{{\text{rss}}} \), for each crystal orientation. Because the stress states were not uniformly uniaxial, the pattern over the orientation space contained some counterintuitive regions that differed starkly with an isostress counterpart. Maps of \( \hat \tau _{{\text{rss}}} \) values for a polycrystalline aggregate illustrated the hard and soft regions of α microstructure.
6.1 Direction
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.
Bragg peaks are typically represented by the Miller indices of associated crystallographic planes, {hkl}; here we use both {hkl} as well as c, which is meant to represent the crystal-relative plane normal.
For the mar345 detector employed in these experiments, a single isotropic offset is sufficient for describing the image distortion.
Acknowledgments
The authors gratefully acknowledge Professor James C. Williams of Ohio State University, for many fruitful conversations related to this project. The research was funded by the United States Office of Naval Research under Contract Nos. N00014-05-1-0505 (MPM, J-SP) and N00014-06-1-0089 (ALP), Dr. Julie Christodoulou, grant officer. The work is based upon research conducted at CHESS, which is supported by the National Science Foundation and the National Institutes of Health/National Institute of General Medical Sciences under Award No. DMR-0225180. The authors also acknowledge Dr. Alexander Kazimirov of CHESS, for his ongoing support of the experimental effort at the A2 station.