Extended Caking Method for Strain Analysis of Polycrystalline Diffraction Debye–Scherrer Rings

Abstract

Polycrystalline diffraction is a robust methodology employed to assess elastic strain within crystalline components. The Extended Caking (exCaking) method represents a progression of this methodology beyond the conventional azimuthal segmentation (Caking) method for the quantification of elastic strains using Debye–Scherrer 2D X-ray diffraction rings. The proposed method is based on the premise that each complete diffraction ring contains comprehensive information about the complete elastic strain variation in the plane normal to the incident beam, which allows for the introduction of a novel algorithm that analyses Debye–Scherrer rings with complete angular variation using ellipse geometry, ensuring accuracy even for small eccentricity values and offering greater accuracy overall. The console application of the exCaking method allows for the accurate analysis of polycrystalline X-ray diffraction data according to the up-to-date rules presented in the project repository. This study presents both numerical and empirical examinations and error analysis to substantiate the method’s reliability and accuracy. A specific validation case study is also presented to analyze the distribution of residual elastic strains in terms of force balance in a Ti-6Al-4V titanium alloy bar plastically deformed by four-point bending.

1. Introduction

The experiment conducted in 1912 by Friedrich and Knipping under the supervision of Laue [1] using a Crookes tube, a single crystal, and a photographic plate resulted in the discovery of X-ray diffraction patterns [2]. This breakthrough in radio crystallography enabled the measurement of X-ray wavelengths and the determination of crystal structures based on observed diffraction patterns. Further studies by Laue demonstrated that the diffraction spots are distributed along conic curves [3]. Later, W.L. Bragg analyzed Laue’s results and proposed the Bragg configuration, which involved the reflection of X-ray beams off crystal planes at a specific angle, known as twice the Bragg angle [4]. This innovative setup, differing from Laue’s, allowed for the measurement of diffracted beams on the same side as the X-ray source, proving Bragg’s theoretical concept and paving the way for advancements in X-ray diffraction studies. Debye and Scherrer also expanded X-ray diffraction studies by exploring diffraction patterns from polycrystalline samples, marking a departure from single-crystal analysis [5]. By irradiating cylindrical samples and measuring diffraction arcs, they applied the Bragg relation to estimate interplanar distances, broadening the applicability of X-ray diffraction beyond single crystals to more prevalent materials.

Synchrotron X-ray scattering [6,7,8] from a polycrystalline aggregate [9] forms a system of coaxial cones of nearly fixed semi-angles, with each cone corresponding to a particular peak with Miller indices ($hkl$) [10,11] that is found on the equivalent 1D powder diffraction pattern [12] for the same material. If a 2D pixelated detector [13] is placed normally to the incident beam, it captures the scattered beam intensity in the perpendicular plane in the form of a concentric pattern that consists of so-called Debye–Scherrer rings [14,15,16,17,18]. For a powder of entirely random orientation, these rings are truly circular [18,19]. If the detector plane is not normal to the incident beam, the conical section produced has the form of an ellipse, with the primary beam passing through one of the foci. If a mechanically loaded or residually stressed polycrystalline specimen [20,21,22,23] that contains internal elastic strains [24,25,26,27] is considered, the ring also becomes distorted into an ellipse, but in contrast to the previous case, this time the incident beam passes through the midpoint between foci and the detector plane is normal to the incident beam, i.e., centrally. Furthermore, the ellipticity parameter due to elastic strain does not deviate from unity by more than 0.02 [28,29] since elastic strain in crystalline metals and ceramics typically does not exceed 1%. Distinguishing between these two types of distortions could present a problem, but luckily the first type (geometric distortion) [30] leads to a radial distance variation of the type $\sin \left(\varphi \right)$ with an azimuthal angle $\varphi$, whilst the second type (strain distortion) [31] leads to an azimuthal variation given by $\sin 2\varphi$ due to the absence of positive/negative directional dependence in the definition of strain. Therefore, the determination of precise experimental geometry (detector inclination) [32] and complete 2D variation in strain requires simultaneous refinement of the entire ring (or system of rings) [33] in a manner not entirely dissimilar to Fourier analysis [15]. The quantification of elastic strains for calculating residual stresses constitutes a significant proportion of polycrystalline diffraction applications [28,34]. This application is built on the premise that mechanical stress induces changes in the crystal lattice interplanar distances [35,36]. The fundamental relation between X-ray scattering geometry and crystal lattice spacing encapsulated in Bragg’s Law [37,38,39] facilitates the discernment of small alterations in lattice spacings [40,41]. Various diffraction techniques [42] that make use of this law include neutron [43,44,45,46,47,48,49,50,51,52], conventional laboratory X-ray [37,53,54,55,56,57], and synchrotron X-ray [58,59,60,61,62,63,64,65,66,67,68]. These observations prompted the use of the entirety of the Debye–Scherrer ring pattern as a single dataset serving as input to ellipse fitting [16,69,70]. The problem is somewhat complicated by the empirical evidence of the additional distortive influences linked to the topographical configuration of the specimen’s surface and variations in pixel dimensions [71].

In various industries, highly accurate measurements of residual stresses are crucial for ensuring the safety, reliability, and optimal performance of materials and structures [72,73,74,75,76]. While determining the distribution of residual stresses provides valuable insights, the precision offered by accurate measurements [77] is paramount in design, manufacturing, and analysis processes [78]. This precision is critical for quality assurance [79,80], as it helps identify and mitigate potential defects or failures. In safety-critical applications like aerospace [81,82], ultra-supercritical power plants [83,84,85,86], and automotive engineering [87], even minor variations in residual stress levels can have significant consequences, making highly accurate measurements essential [88]. Additionally, precise data aid in the optimization of manufacturing processes [89] and the validation of numerical models [90] and contribute to materials research and development. Accurate measurements serve as a cornerstone for cross-verifying results, standardizing testing procedures, and enhancing overall consistency in the industry [91]. Ultimately, the pursuit of highly accurate measurements is crucial in advancing the understanding and control of residual stresses [92,93], impacting the entire lifecycle of materials and structures [94]. The Caking method introduced by Korsunsky et al. [95] emerged as a practical and viable strategy involving the averaging of deviations of the Debye–Scherrer rings from their reference morphology [96] for a number of chosen azimuthal angular ranges to satisfy the requirement for quantifying residual stresses with high accuracy.

The Caking method has proved its value and reliability over the last three decades [97,98,99,100,101,102,103,104,105,106]. However, it relies on the conversion of 2D detector data via radial–azimuthal binning for chosen sectors separately [107,108] without attempting a full fitting of the ellipse due to its extremely small eccentricity [101,109]. Once the radial positions of the rings are known, this data may be interpreted in connection with the full 2D elastic strain variation as an additional analysis step. In contrast, the present research introduces a novel algorithm that performs a complete angular variation analysis of Debye–Scherrer rings by exploiting the geometric properties of an ellipse, maintaining accuracy even for small eccentricity values. The advantages of this approach include greater accuracy achieved via a faster, single-step analysis. In this study, numerical analyses demonstrate the error reduction achieved through the employment of the newly proposed algorithm. For validation, a case study was conducted using synchrotron X-ray diffraction data obtained from a Ti-6Al-4V alloy subjected to elastic-plastic four-point bending. This examination illustrates the accuracy and reliability of the exCaking method compared to conventional Caking utilizing a custom MATLAB code and GSAS-II ellipse-fitting analysis. This presentation also includes a comparison of force balances that validates the improvement in accuracy when compared to other methods.

2. Methodology

The Caking method evaluates elastic strains along a chosen direction through the determination of radial displacement of the peak center in relation to a reference position. This methodology involves polar rebinning of pixel intensity data into angular bins relative to the pattern center within an azimuthal angle range of $\varphi$ at a transformation angle of ${\varphi }_T$, as depicted in Figure 1a. Conversely, the exCaking method, which is formulated as an extension of the Caking method, utilizes the trigonometric averaging of the $x\prime$- and $y\prime$-components of distance from the ring center within the azimuthal span range of $2\pi$ at a transformation angle of ${\varphi }_T$, as depicted in Figure 1b. It is important to note that the azimuthal angle step size necessitates adjustment in accordance with the observed data scatter. This formulation allows extraction of the information about the elastic strain components from the whole ring in spite of a limited angular range.

The reference position, which is the center of the Debye–Scherrer 2D X-ray diffraction rings, is determined using calibration rings that are fitted into a circle. Subsequent to the determination of the coordinates of the center of the beam, the radial distance of peaks from the center, denoted as $D^{\varphi }$ at angles denoted as $\varphi$ varying in a range of 360 degrees with a predetermined angle step size, is determined through Gaussian peak fitting, as depicted in Figure 2a. Finally, trigonometric averages of the $x\prime$- and $y\prime$-components of the radial distance from the predetermined ring center, denoted as $D_{x\prime }^{\prime }$ and $D_{y\prime }^{\prime }$, are computed at a transformation angle of ${\varphi }_T$ utilizing Equations (1) and (2).

$$D_{x\prime }^{\prime }=\underset{0}{\overset{2\pi }{\int }}{D}^{\varphi }\left|\cos \varphi \right|d\varphi /\underset{0}{\overset{2\pi }{\int }}\left|\cos \varphi \right|d\varphi =\underset{0}{\overset{2\pi }{\int }}{D}_{x\prime }^{\varphi }d\varphi /\underset{0}{\overset{2\pi }{\int }}\left|\cos \varphi \right|d\varphi$$

(1)

$$D_{y\prime }^{\prime }=\underset{0}{\overset{2\pi }{\int }}{D}^{\varphi }\left|\sin \varphi \right|d\varphi /\underset{0}{\overset{2\pi }{\int }}\left|\sin \varphi \right|d\varphi =\underset{0}{\overset{2\pi }{\int }}{D}_{y\prime }^{\varphi }d\varphi /\underset{0}{\overset{2\pi }{\int }}\left|\sin \varphi \right|d\varphi$$

(2)

Trigonometric averaging is applied across the entire range of 360 degrees, encompassing the complete angular variation in the Debye–Scherrer ring. This comprehensive angular coverage ensures that data from all points around the ring contribute to the averaging calculations. As a result, any error in the determination of the coordinates of the ring’s center is rendered negligible in its influence on the final averaged values. This phenomenon can be understood from a mathematical perspective. Trigonometric averaging involves integrating the radial distances from the ring’s center over the full angular range. Since the integration process considers all points evenly distributed around the ring, errors in individual coordinate measurements are effectively distributed and balanced out over the entire range. Consequently, any inaccuracies in determining the center coordinates have a minimal impact on the overall averaging calculations, leading to robust and reliable results.

Based on empirical analysis utilizing the semi-minor and semi-major axes of the artificially strained ring illustrated in Figure 2b, trigonometric averaging values of distances corresponding to the transformed coordinate axes are updated to $D_{x\prime }$ and $D_{y\prime },$ as outlined in Equations (3) and (4).

$$D_{x\prime }=2D_{x\prime }^{\prime }-D_{y\prime }^{\prime }$$

(3)

$$D_{y\prime }=2D_{y\prime }^{\prime }-D_{x\prime }^{\prime }$$

(4)

For monochromatic (fixed wavelength) diffraction, Bragg’s Law leads to the relation between elastic strain (relative lattice spacing change) and the scattering angle increment given by $\epsilon =\delta d/d=-\cot \theta \delta \theta$. For the sample–detector distance $L$, the distance from the ring center is set by $\tan 2\theta =D/L$. The variation in the scattering angle, $2\delta \theta$, is linked to the change in the distance from the ring center, $\delta D$, as $2\delta \theta /{\sin }^{2}2\theta =\delta D/L$. Hence, $\tan 2\theta \delta D/D=2\delta \theta /{\sin }^{2}2\theta$, and $\epsilon =(-\cot \theta {\sin }^4\theta \left)\right(\delta D/D)$. The geometric factor that appears above remains remarkably close to unity for small scattering angles. For the experimental parameters considered in the present study, $\epsilon =0.993 \delta D/D$, i.e., <1% difference. Furthermore, the variation in the geometry factor as a function of the Bragg angle is exemplified in Figure 2c. In the limit of $\theta \to 0,$ the multiplying factor converges to unity, so for small scattering angles, the following approximation is found to hold: $\epsilon =-\delta D/D$. Finally, the quantification of the elastic strain’s $x\prime x\prime$- and $y\prime y\prime$-components at a transformation angle of ${\varphi }_T$ hinges upon the utilization of Equation (5), which employs the stress-free reference distance from the center, denoted as $D_0$.

$${\epsilon }_{x\prime x\prime ,y\prime y\prime }\cong \left(D_0-D_{x\prime ,y\prime }\right)/D_0$$

(5)

Numerical analyses were performed to determine elastic strains as a function of azimuthal angles over a range from 0 to 2$\pi$, with an azimuthal angle step size of $\pi$/180. The reference unstrained ring was determined to have a radius of 100 pixels. For the ‘artificially strained’ ring, the semi-minor axis of the ellipse was set to 99 pixels, while the semi-major axis was set to 102 pixels. Numerical analysis involved calculating the direct elastic strain components within a transformation range from 0 to 2$\pi$, with a step size of $\pi$/180 for both the exCaking and Caking methods. Concurrently, an azimuthal step size of $\pi$/180 was determined for the polar rebinning of the Caking method and for the trigonometric averaging of the exCaking method. The Caking method was applied according to the guidelines outlined in the study that introduced this method [95], and the Caking range was set to 15$\pi$/180.

Debye–Scherrer data were collected from the middle segment of this titanium bar around the symmetry axis of four-point bending over a grid of 9 columns and 51 rows, with step sizes of 1 and 0.1 mm along the $x$- and $y$-axes, respectively, as illustrated in Figure 3b. The baseline state of the titanium bar’s ring structure was established through the determination of the mean $D_{x\prime }$ and $D_{y\prime }$ values calculated at an azimuthal angle of zero for all experimental measurements. A synchrotron X-ray beam with a photon energy of 72 keV was collimated to a beam spot size of 76 × 115 µm and a 2.856-degree diffraction angle. Diffraction patterns were recorded using the 2D detector Pilatus P3-2M (Dectris, Switzerland), with a matrix of 1679 × 1475 pixels.

The experimental assessment of elastic strain quantification methods followed the same transformation and azimuthal angle step sizes as determined in the numerical analysis using the selected ring illustrated in Figure 3a. For this purpose, a Debye–Scherrer ring with a Miller index of ($102$), indicated by the red arrows in Figure 3a, was obtained using synchrotron X-ray diffraction scanning of a four-point bent titanium alloy specimen and selected for experimental analysis of the Caking and exCaking methods. The selection of this ring was based on the sensitivity of lattice planes with low Miller indices to small changes in lattice spacing, making them suitable for detecting subtle strains within the crystal lattice. The incomplete and missing data on the ring were omitted after identifying them as outliers using standard deviation-based error analysis according to the three-sigma rule.

The bending of this specimen was performed to induce plastic deformation in regions close to the faces normal to the $y$-axis while keeping the central region free of plastic deformation. The resulting $xx$- and $yy$-components of elastic strain in the Cartesian coordinate system of both techniques within the selected ring were fitted against the transformation function formulated in Equation (6) and compared with calculations at transformation angles.

$$\epsilon ({\varphi }_T)={\displaystyle \frac{{\epsilon }_{xx}+{\epsilon }_{yy}}{2}}+{\displaystyle \frac{{\epsilon }_{xx}-{\epsilon }_{yy}}{2}}\cos ({2\varphi }_T)$$

(6)

Residual stresses distributed over the measurement grid on the four-point bent titanium alloy bar were calculated using the planar stress assumption in the Cartesian coordinate system using Equation (7), with Young’s modulus ($E$) and Poisson’s ratio ($v$) set to 115 GPa and 0.34 [110].

$${\sigma }_{xx,yy}={\displaystyle \frac{E\left({\epsilon }_{xx,yy}+v{\epsilon }_{yy,xx}\right)}{1-v^2}}$$

(7)

The console application of the exCaking method was developed using MATLAB R2022a, a proprietary, multi-paradigm programming language and numeric computing environment developed by MathWorks. This application enables the accurate determination of trigonometric averaging values of distances corresponding to the transformed coordinate axes in the updated form. The flowchart of the application given in Figure 2d begins with the calibration step, the aim of which is to determine the center of the Debye–Scherrer ring. This stage is recommended to be completed using a calibration image of Debye–Scherrer rings obtained from the same setup of polycrystalline X-ray diffraction scans of the specimen. Following the calibration step, the user selects the rings to be analyzed from any image of the Debye–Scherrer ring data collected from the specimen. Finally, the solver of the console application is initiated, and the results are stored in plain text formats. The project repository contains the up-to-date rules of the exCaking console application.

3. Results

Polar representations depicted in Figure 4a demonstrate that the exCaking method yields direct elastic strain components for artificially strained rings with an error not exceeding 0.04%. By contrast, quantification via the Caking method results in errors greater than 7.3% for the $yy$-component and exceeding 3.5% for the $xx$-component within the same strained ring, as depicted in Figure 4b. These observations illustrate that the error percentage of the Caking method increases with diminishing magnitudes. Additionally, the highest error is noted in the $xx$- and $yy$-components of elastic strain in the Cartesian coordinate system aligned with the relative lattice rotation, while the error diminishes at intermediate angles.

Evident from the outcomes depicted in Figure 4c, the Caking method displayed a distribution characterized by noise, whereas the exCaking method exhibited a notably smoother polar distribution. The fitting mean error for the exCaking method was substantially lower in comparison to the Caking method’s solution, depicted in Figure 4d. In congruence with the numerical analysis, while the exCaking fitting error was anticipated to be under 0.04%, practical errors stemming from data collection procedures introduced deviations in the determination of distances from the center positions on the ring, thus amplifying the fitting error. Consequently, the mean error, quantified at 1.637% for the exCaking method, remains variable across different Debye–Scherrer ring data; nonetheless, the Caking method consistently yields a notably higher mean error of 13.071% in comparison to the numerical analysis-derived errors of the $xx$- and $yy$-components of elastic strain in the Cartesian coordinate system. Although both methods exhibit analogous distribution trends, their magnitudes diverge, culminating in the deduction that the exCaking method offers a substantively more accurate and reliable measure of elastic strain in contrast to the Caking method.

This case study, which involved a comprehensive grid, facilitated an assessment of residual elastic strain distribution in a titanium alloy bar after it underwent plastic four-point bending using both the exCaking and Caking methods. Elastic strains averaged across grid rows, as depicted in Figure 4e, reveal that the exCaking outcomes exhibit a smoother distribution closely resembling the theoretically anticipated distribution of bending residual elastic strains. By contrast, the Caking method, as depicted in Figure 4f, yields a discordant distribution characterized by noise and distortion attributable to errors elucidated via numerical and experimental analyses. Subsequent quantitative examination of the same dataset utilizing GSAS-II crystallography data analysis software [111] also involved ellipse fitting.

Eigenstrain reconstruction of residual elastic strains in a similar specimen of the same alloy [112] further corroborated the distribution of residual elastic strain due to elastic-plastic four-point bending. This investigation also revealed that the averages of the $xx$-components of bending residual stresses, which were distributed within the diffraction map grid, were determined to be −0.199, −13.404, and 2.231 MPa for the exCaking, Caking, and ellipse-fitting solutions, respectively. The relationship between force balance and the accuracy of strain quantification methods is integral to understanding the reliability of residual stress assessments in materials. Force balance, representing the equilibrium of internal stresses within a specimen, is directly influenced by the accuracy of residual elastic strain measurements. This study demonstrates that the exCaking method consistently provides more accurate results compared to the traditional Caking method. This higher accuracy translates to a more precise determination of residual stresses, thereby impacting the overall force balance within the material. Consequently, it can be concluded that the exCaking method provides the most reliable solution in terms of force balance. The high deviation from force balance achieved by the Caking method can be attributed to the elevated error range observed in the numerical experiments, while the low deviation from force balance of the exCaking method is consistent with the low error range presented in the polar representations shown in Figure 4a–d.

The exCaking method represents a significant advancement in polycrystalline diffraction analysis, particularly in the quantification of elastic strain within crystalline materials. This method extends beyond the conventional Caking approach by utilizing complete angular variation analysis of Debye–Scherrer rings, leveraging ellipse geometry properties for accurate strain determination. The exCaking algorithm ensures precision even for small eccentricity values and offers a faster, single-step analysis, enhancing overall accuracy. The numerical and experimental examinations presented in this study validated the reliability and accuracy of the exCaking method. Specifically, the numerical analyses demonstrated a reduction in error compared to the Caking method, showcasing the efficacy of the new algorithm. Additionally, a validation case study conducted using a titanium alloy bar subjected to elastic-plastic four-point bending confirmed the accuracy and reliability of the exCaking method. The comparison between the exCaking and Caking methods revealed that exCaking produced smoother polar distributions and lower fitting errors, indicating superior performance. The accuracy of the exCaking method underscores the notion that individual points across Debye–Scherrer rings contain valuable data about elastic strain components in polycrystalline materials. By extracting this information through trigonometric averaging, the exCaking method achieved increased accuracy in strain quantification. Experimental analyses further validated the method’s effectiveness, highlighting its potential for various applications requiring precise determination of residual elastic strains and stresses. Overall, the exCaking method offers a more coherent and accurate approach to strain analysis compared to conventional methods, making it a preferred choice in scenarios demanding high accuracy and reliability.

4. Conclusions

A comparative analysis of the exCaking method was undertaken through both numerical and experimental analyses. The numerical investigations revealed that the exCaking method led to negligible errors in quantifying the $x\prime x\prime$- and $y\prime y\prime$-components of elastic strains at transformation angles of ${\varphi }_T$ within the artificially strained ring. By stark contrast, the Caking method resulted in quantifications with a high error within the same ring. The effectiveness of the exCaking method substantiates the assertion that individual points across Debye–Scherrer rings harbour valuable data regarding the components of elastic strain within polycrystalline materials. Consequently, extracting this data through trigonometric averaging yields increased accuracy. Experimental analysis corroborated the numerical findings, as both methods were evaluated under the same conditions. The exCaking method exhibited smoother polar distributions, consistently outperforming the significantly higher mean error associated with the Caking method. The subsequent application of both methods to a titanium alloy bar subjected to four-point bending confirmed the exCaking method’s capacity to produce a more coherent and accurate distribution of residual elastic strains compared to the Caking and ellipse-fitting methods. The results of the case study also demonstrate that all three methods were successful in determining the expected distribution of bending residual stress but with varying deviations from force balance. Accordingly, the exCaking method and its console application (see Supplementary Materials) in the Supplementary Materials should be preferred in cases where an accurate determination of residual elastic strains and residual stresses is needed.