Evaluation of Measurement Uncertainty for Residual Stress Measurements using X-Ray Diffraction Chan Shi Ying a, Pooja Chaturvedi a, Dennise Tanoko Ardi a, Keegan Lau a, More info about this article: http://www.ndt.net/?id=22177 a Advanced Remanufacturing and Technology Centre, Agency for Science, Technology and Research (A*STAR), Singapore E-mail: chanshy@artc.a-star.edu.sg Abstract The study of residual stress is important in the science and engineering field due to its high relevance to the performance of products. For uncontrolled residual stresses, they can have an adverse effect on product safety and quality, whereas, compressive residual stresses are able to improve the fatigue performance of a product. Given its spatial and volumetric resolution, X-ray Diffraction (XRD) is able to adequately characterise and quantify the residual stress distribution of any polycrystalline component at critical locations. The quality of this measurement result, its precision and accuracy are, however, defined by measurement uncertainty. In the current study, the individual contribution by various factors and measuring equipment were evaluated to compute the final meansurement uncertainty. Environmental factors like temperature and humidity were omitted given that the residual stress measurements were performed in a highly controlled environment (20±2 C and <70%RH). It was observed that resolution of the XRD equipment used and test procedures such as collimator distance from the measured surface, peak fitting parameters and human bias are the dominating factors. Overall, time and effort were devoted to the quantification of measurement uncertainty to increase the precision and accuracy of residual stress measurements for the purpose of designing and manufacturing engineering components with increased fatigue strength and fatigue life. Thus, leading to enhanced product safety, quality and productivity. Keywords: Residual stresses, X-ray Diffraction, measurement uncertainty, precision and accuracy 1 Introduction XRD measurement techniques have been used to determine the near-surface stress state of finite crystalline materials for many decades, and significant advances have been made to both hardware technologies and analysis methods 1. The use of laboratory X-ray stress analyser, Xtress 3000 G3, which utilises two symmetrically positioned detectors, is common in laboratories and field applications. In laboratory conditions, these measurements are convenient and can be performed on hard-to-reach areas because the G3 s design, without a tilting arc, leaves clearance under the goniometer to accommodate
complex or tall features of test samples. In addition, the G3 is able to automatically adjust the distance between the goniometer and the measurement point on the test sample. The evaluation of uncertainty is essential to the understanding of various contributing factors, finally leading to making accurate measurements. High precision and accuracy of the measured stress must be balanced with other important criteria, such as test sample, XRD equipment as well as test procedure. Therefore, this paper presents an analysis of the uncertainties associated with the state-of-the-art commercial laboratory X-ray equipment. 2 Material Preparation In this study, a nickel-based super alloy (IN718) was chosen for detailed analysis. It is a highly oxidation-corrosion-resistant polycrystalline material, well suited to work in environments subjected to extreme pressure and heat 2. It is also able to retain strength over a wide temperature range as a result of precipitation hardening 3. Additionally, the IN718 test sample was shot-peened to induce a certain amount of compressive residual stress. The requirement for surface roughness of the test sample is strictly 5 microns, so as to reduce any possibility of uncertainty caused by sample surface roughness. Thus dimples embedded on the IN718 test sample created by shot peening were smoothened out by electropolishing, a chemical surface-finishing technique 4. Care was taken while cleaning the surface of IN718 sample by lint-free paper and ethanol to remove any tarnished oxide layer as shown in Figure 1 where X indicates the measurement location. Figure 1: Shot-peened IN718 test sample with electro-polished surface.
3 Measurement Uncertainty Analysis The generic procedure for measurement uncertainty analysis is listed in Figure 2. Figure 2: Generic uncertainty analysis procedure Based on ISO-GUM 6, the first step towards quantifying uncertainty factors requires developing the XRD measurement process in adequate detail, followed by identifying every uncertainty factor associated with each step of the measurement process. This can be from varied sources such as technical manuals and specifications, calibration reports as well as conducting in-house Repeatability and Reproducibility (R&R) study. The uncertainty factors identified from test sample, equipment and procedure during the measurement process are classified as or uncertainty, and they are to be evaluated accordingly with consideration of effectiveness and practicality. uncertainty and uncertainty presents the two different ways of data collection and evaluation. uncertainty is derived from a series of repeated random observations, such as via a Repeatability and Reproducibility (R&R) Study (to be further elaborated in Section 3.1), where treatment of data is based on any valid statistical method 5. On the other hand, uncertainty evaluation is usually based on scientific judgement using all of the relevant information available, which may include: previous measurement data, experience with, or general knowledge of, the behaviour and properties of relevant materials and instruments, manufacturer s specifications, data provided in calibrartion and other reports, and uncertainties assigned to reference data retrieved from handbooks 6
Table 1 lists the uncertainty sources associated with XRD measurement on Xtress 3000 G3 and their uncertainty type. Contribution of uncertainty from temperature and humidity was eliminated as the test environment was maintained strictly at 20±2 C and <70%RH. Uncertainty Source Alignment of instrument (R&R) Collimator distance from surface (R&R) Peak position determination method (R&R) Peak fitting paratmeters (R&R) Human bias (R&R) Detector step size (fixed) Physical constant E Physical constant v Length of counting time Microstructure features (texture, grain size) Stress gradient Surface condition (roughness) Stress-free powder stress tolerance Resolution Temperature Goniometer error Goodness of fit Uncertainty Type Table 1: Identified uncertainty factors and their uncertainty type A mathematical model was developed for the calculation of the combined standard uncertainty of the measurement as a function of standard uncertainties of all identified uncertainty factors. The uncertainties were expressed as a relative uncertainty or absolute uncertainty, depending on the mathematical relationship. Figure 3 shows the procedure for development of a mathematical model for the evaluation of the combined uncertainty as well as the expanded uncertainty which is the desired outcome of this study.
Figure 3: Procedure for development of mathematical model for the evaluation of the combined uncertainty and the expanded uncertainty (Adapted from Evaluation of Uncertainty in Dimensional Measurements by National Metrology Centre, Singapore) After the uncertainty modelling (together with identification of uncertainty factor) as aforementioned, quantification of individual factors, calculation of combined standard uncertainty and calculation of expanded uncertainty (95% confidence interval) ensued. 3.1 Repeatability and Reproducibility Study The R&R Study forms the basis for uncertainty assessement for uncertainty. Using IN718 test sample (as described in Section 2), a R&R Study, representative of the measurement process in actual conditions, was conducted to determine the amount of variation in the measurement system arising from human bias and test procedure. Three operators were employed to perform ten unique measurements on the same IN718 test sample, using the three available G3 equipment - equipment 1, 2 and 3. A total of 90 stress measurements were
collected and this data, along with other uncertainty sources like alignment of the XRD equipment, collimator distance from test sample surface, peak position determination method, peak fitting parameters and goodness of peak fitting, were subsequently evaluated as uncertainty. 3.2 Calculation of Combined Standard Uncertainty It is usually assumed that the uncertainty factors identified as can be modelled as normal distribution (Figure 4). The calculation of combined standard uncertainty began with reducing the Type A uncertainty factors, listed in Table 1, to their standard deviation (S.D.) equivalents. In the normal distribution, u is the standard deviation equivalent and a is the value of the input quantity. The width of the probability distribution (±a) is indicative of the interval of possible values. Figure 4: Normal distribution Table 2 presents the identified uncertainty factors (Table 1) reduced to their their respective S.D. equivalent determined using normal distribution function (Figure 4). Uncertainty Source Uncertainty Type 1 Std devi Alignment of instrument (R&R) - d Collimator distance from surface (R&R) - d Peak position determination method (R&R) - d 18.67 Peak fitting paratmeters (R&R) - d Human bias (R&R) - d Detector step size (fixed) - d 0 Physical constant E - E 0 Physical constant v - v 0 Length of counting time - d 0 Microstructure features (texture, grain size) - d 0 Stress gradient - d 0 Surface condition (roughness) - d 0 Stress-free powder stress tolerance - d 0 Resolution - d 0.1 Temperature 2 Goniometer error 0 Goodness of fit 0 Table 2: Calculated standard deviation of each uncertainty factor
Subsequently, the S.D. equivalents were combined to estimate the combined standard uncertainty. To accomplish this, the Root Sum of Squares (RSS) method was used to mathematically combine the uncertainty factors in quadrature. It involved squaring the value of each uncertainty factor, followed by addition of all these squared values to calculate the sum (i.e. the sum of squares). This summed value wass, again, subjected to square-root (i.e. the root sum of squares) to give the combined standard uncertainty, (1). When the uncertainty factors are being combined, the probability distributions are also combined. According to the Central Limit Theorem, the sum of all independent uncertainty factors will approach a normal distribution regardless of individual factor s distribution. Thus, the probability distribution of the combined standard uncertainty is normal. = + + + (1) 3.3 Calculation of Expanded Uncertainty The final step towards the determination of measurement uncertainty involves expanding the combined standard uncertainty to an acceptable level of confidence using a suitable coverage factor, k. k is chosen from the T-test table (Appendix 1) with an effective degrees of freedom,, obtained from the Welch Satterthwaite equation (2) = 4 4 = (2) In this study, for practical purpose and the approximate nature of the uncertainty evaluation process, a coverage factor of k=2 was adopted which defined an interval having a confidence level of approximately 95%. (3) shows that expanded uncertainty can be achieved by the multiplication of the chosen k value (i.e. k=2) and the combined standard uncertainty,. U = k (3) 4 Conclusion The quality of the measurement result, its accuracy, is characterized by measurement uncertainty. All measurements are subject to uncertainty and a measurement result is complete only when accompanied by the associated uncertainty. In the XRD measurement technique, its accuracy is determined by many factors, such as instrumental parameters (beam centre alignment and calibration), measurement
parameters (counting time and tilt angles), sample conditions (texture and grain size) and physical constants (Poisson s ratio and Young s modulus). These factors, otherwise known as uncertainty sources, are evaluated as according to their suggested methods as listed in Table 3. Type Uncertainty Source Evaluation Method A B Repeatability of measurement instrument (e.g. Xtress 3000 G3) Uncertainty of measurement instrument, due to calibration uncertainty, drift, resolution, etc Uncertainty of measurement condition, without explicit mathematical expression Measure the quantity repeatedly and analyze the result statistically Refer to the specifications of the measurement instrument dor relevant uncertainty data Conduct sensitivity test and estimate the uncertainty Table 3: and B uncertainty and recommended evaluation Following the procedures discussed in the earlier sections of this paper (Figure3), an Uncertainty Budget summarising the possible uncertainty at each analysis step (Table 4) was generated based on the present work. The overall uncertainty is a combination of the contributions related to the uncertainty sources (column 2 of Table 4). It is concluded that that test procedures involving alignment of the XRD equipment, collimator distance from test sample surface, peak position determination method, peak fitting parameters, goodness of peak fitting as well as human bias from the GR&R Study have the greatest combined degree of uncertainty in XRD measurements using the Xtress 3000 G3. These are uncertainty factors typically customizable by the operators. Therefore the effect of different operators is incorporated in this repeatability test so that human measurement variation, also known as human bias. The remained uncertainty factors were rendered negligible or insignificant in the study as their S.D equivalents approximate to zero.
Table 4: Uncertainty Budget for XRD Measurement Process
6 Future Work With this knowledge of measurement uncertainty, future work can extended to reduce or eliminate the identified uncertainty factors in the stress measurement process. The optimisation of process parameters and the study of the propagation of the effects of uncertainties in the case of one-dimensional XRD measurement can be a potential field of research. Stress measurement values, often than not, differ from the known stress applied, and only a narrow range of measurement conditions have been tested. It is suggested to perform multiple measurements over a set of experimental conditions to identify external and/or sampling uncertainties. For instance, future investigations of measurement precision and accuracy can be performed in an environment that resembles field or production line conditions. In addition, more material types, besides IN718, should be explored as different grain sizes and texturing of different materials can play a critical role in the uncertainty of measurement. Furthermore, the calculation of measurement uncertainty, in this study, is applicable only to onedimensional XRD technique. Hence it can extend to two-dimensional XRD tecnhique, which uses area detectors, in the near future which might then involve more elements to consider. Acknowledgements The authors would like to thank Advanced Remanufacturing and Technology Centre (ARTC) for providing the resources, facilities and financial support. I would like to express my special thanks to Dr. Cheng Fang for his valuable inputs towards the identification of uncertainty factors. References [1] Noyan, I. C. & Cohen, J. B. (1987). Residual Stress. New York: Springer-Verlag [2] Bayrak, Özgü & Vangolu, Yenal & Yetim, A.F. & çelik, Ayhan. (2017). CORROSION BEHAVIOR OF PLASMA SURFACE OXIDIZED INCONEL 718 SUPER ALLOY. [3] K., & D. (august 2015). Strengthening of Forged Inconel Superalloy by Age Hardening Heat Treatment. - International Journal of Innovative Science, Engineering & Technology, 2(8). [4] Passivation vs. Electropolishing Stainless Steel Finishing Process. (n.d.). Retrieved from http://www.ableelectropolishing.com/resources/passivation-vs-electropolishing/ [5] Hogan, R. (2017, October 09). and Uncertainty: Evaluating Uncertainty Components. Retrieved from http://www.isobudgets.com/type-a-and-type-b-uncertainty/
[6] JCGM 100:2008 GUM 1995 with minor corrections - Evaluation of measurement data Guide to the expression of uncertainty in measurement [7] ISO/IEC 17025:2005 - General requirements for the competence of testing and calibration laboratories [8] Williams, J. H. (2016). Quantifying measurement: the tyranny of numbers. San Rafael, CA: Morgan et Claypool [7]
Appendix T-test Table