STANDARD OF THE COSa METHOD FOR X-RAY
STRESS MEASUREMENT USING
TWO-DIMENSIONAL DETECTOR

Yoshihisa Sakaida, Kenji Suzuki, Keisuke Tanaka

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Yoshihisa Sakaida, Kenji Suzuki, Keisuke Tanaka. STANDARD OF THE COSa METHOD FOR X-RAY STRESS MEASUREMENT USING TWO-DIMENSIONAL DETECTOR. ICRS 11 - The 11th International Conference of Residual Stresses, SF2M; IJL, Mar 2022, Nancy, France. hal-04015069

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# STANDARD OF THE COSα METHOD FOR X-RAY STRESS MEASUREMENT USING TWO-DIMENSIONAL DETECTOR

Yoshihisa Sakaidaᵃ, *, Kenji Suzukiᵇ, Keisuke Tanakaᶜ

## ABSTRACT

In order to standardize the cosα method for X-ray stress measurement using a two-dimensional detector, the subcommittee was set up under the “Committee on X-ray Study for Mechanical Behavior of Materials” of the Society of Materials Science, Japan (JSMS) in 2015. In the interlaboratory collaboration, a round robin test using carbon steel with 0.46 mass% C was carried out to formulate a standard method for measuring stresses in the ferritic steels under the plane stress condition. Thirteen members from academia and industry participated in the round robin test. Based on the round robin test results, the Japanese standard for X-ray stress measurement by the cosα method for ferritic steels was published in 2020. The 211 diffraction of α-Fe by Cr-Kα radiation is adopted for stress measurements of ferritic steels. The standard recommends the X-ray incident angle ψ₀ = 35° and the X-ray elastic constant E' = E<sub>X</sub> / (1 + ν<sub>X</sub>) = 175 GPa (E<sub>X</sub>: Young's modulus, ν<sub>X</sub>: Poisson's ratio). The standard also offers a concise description of the underlying measurement principle and the protocol for accurate measurements.

**Keywords:** standard method; X-ray stress measurement; the cosα method; Debye-Scherrer ring; two-dimensional detector

## 1. Introduction

The X-ray diffraction technique is recognized as one of the most powerful nondestructive tools for measuring residual stresses in polycrystalline materials, and has been widely used in-field as well as in laboratory investigations. The sin²ψ method is most commonly used, in which the stress is determined from the slope of the plot of the diffraction angle or lattice strain versus sin²ψ, where the diffraction angles are measured at several tilt angles ψ. In recent years, the cosα method has attracted attention as an alternative approach to measuring the stress much faster without loss of accuracy [1, 2, 3]. The cosα method utilizes the entire Debye-Scherrer (D-S) ring recorded by a two-dimensional detector or an area detector under a single exposure of X-rays at an oblique angle. The stress is determined from the slope of the plot of a strain parameter versus cosα, where α is a rotation angle of the D-S ring.

The principle of the cosα method was first proposed in 1978 using X-ray films as detectors [4], and later imaging plates were introduced for the same purpose [5]. In 2012, a first

<footnote>ᵃ Department of Mechanical Engineering, Shizuoka University, Naka-ku, Hamamatsu, Japan</footnote>

<footnote>ᵇ Faculty of Education, Niigata University, Nishi-ku, Niigata, Japan</footnote>

<footnote>ᶜ Professor Emeritus of Nagoya University, Chikusa-ku, Nagoya, Japan</footnote>

<footnote>* Corresponding author. sakaida.yoshihisa@shizuoka.ac.jp</footnote>
commercial stress analyzer, μ-X360n, was produced by Pulstec Industrial Co. Ltd. The simple optical system adopted by the cosα method makes stress analyzers portable, smaller, lighter, and handier for on-site measurements [6]. The time required for one stress measurement is shortened to as little as 60 sec. Nowadays, the cosα method has been adopted by more than 300 facilities worldwide. However, because of the wide-ranging applications in various engineering fields, the standardization of the cosα method became necessary to guarantee the reliability of the measured values.

In 2015, the subcommittee was set up for standardization of the cosα method under the “Committee on X-ray Study for Mechanical Behavior of Materials” of the Society of Materials Science, Japan (JSMS). Thirteen subcommittee members from academia and industry participated in a round robin test. This interlaboratory collaboration was conducted using the round robin specimen of carbon steel with 0.46 mass% C in order to determine the optimal procedure for accurate stress measurements. Based on the round robin test results, the Japanese standard for the X-ray stress measurement by the cosα method was published in 2020 [7, 8].

In this article, the essence of the standardized method is presented with several explanatory comments, and the results of the round robin test is included as Appendix.

## 2. Standardized method

### 2.1. Scope

The cosα method using a two-dimensional detector is a method for measuring the surface stress of polycrystalline materials by X-ray diffraction. The standard offers a concise description of the underlying measurement principle, protocol for accurate stress measurements in ferritic steels under the plane stress condition, and several best-practice guidelines for making measurements.

### 2.2. Principle of the cosα method

Figure 1 shows a D-S ring obtained from a single exposure of X-rays with the incident angle $\psi_0$. Figure 2 illustrates four strains to be measured in the D-S ring. The strain $\epsilon_α$ at the rotation angle $α$ is determined from the radius $r_α$ of the D-S ring as follows:

$$ \epsilon_{\alpha} = -(\theta_{\alpha} - \theta_{0}) \cdot \frac{\pi}{180} \cot \theta_{0} = \frac{\cos^2 2\theta_{0}}{2L \tan \theta_{0}} (r_{\alpha} - r_{0}), \qquad (1) $$

where $\theta_α$ is the diffraction angle at $α$, $\theta_0$ is the diffraction angle and the $r_0$ is the radius of a stress-free sample, and $L$ is the distance between the detector and the specimen surface.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Debye-Scherrer ring of the specimen.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2. Geometry of the Debye-Scherrer ring and the specimen, and the determination of four lattice strains in the cosα method.</figcaption></figure>
By assuming Hooke's law under the plane stress condition, the lattice strain $\epsilon_{\alpha}$ is related to in-plane normal stresses, $\sigma_x$, $\sigma_y$, and shear stress, $\tau_{xy}$, by the following formula:

$$ \epsilon_{\alpha} = \frac{1 + \nu_X}{E_X} \left[ \begin{aligned} & (\cos \eta \sin \psi_0 - \sin \eta \cos \psi_0 \cos \alpha)^2 \sigma_x + \sin^2 \eta \sin^2 \alpha \cdot \sigma_y \\ & + 2 \sin \eta \sin \alpha (\cos \eta \sin \psi_0 - \sin \eta \cos \psi_0 \cos \alpha) \cdot \tau_{xy} \end{aligned} \right] - \frac{\nu_X}{E_X} (\sigma_x + \sigma_y), \quad (2) $$

where $\eta = 90 - \theta_{\alpha}$, $E_X$ and $\nu_X$ are the X-ray values of Young's modulus and Poisson's ratio, respectively.

In the $\cos \alpha$ method, a special technique is adopted to determine the stress values in order to minimize potential experimental errors [4]. As shown in Fig. 2, four strains, $\epsilon_{\alpha}$, $\epsilon_{\pi+\alpha}$, $\epsilon_{-\alpha}$, $\epsilon_{\pi-\alpha}$, are determined for each $\alpha$ from the radius of the D-S ring using Eq. (1). Then, differences ($\epsilon_{\alpha} - \epsilon_{\pi+\alpha}$) and ($\epsilon_{-\alpha} - \epsilon_{\pi-\alpha}$) are used to calculate the following strain parameters $\epsilon_{\alpha1}$:

$$ \epsilon_{\alpha1} = \frac{(\epsilon_{\alpha} - \epsilon_{\pi+\alpha}) + (\epsilon_{-\alpha} - \epsilon_{\pi-\alpha})}{2}. \quad (3) $$

Substituting Eq. (2) into Eq. (3), the strain parameter is found to be related to the normal stress, $\sigma_x$, as follows:

$$ \epsilon_{\alpha1} = -\frac{1 + \nu_X}{E_X} \sin 2\eta \sin 2\psi_0 \cos \alpha \cdot \sigma_x \quad (4) $$

By partial differentiation of the above equation with $\cos \alpha$, the normal stress $\sigma_x$ can be expressed as follows:

$$ \sigma_x = -\frac{E_X}{1 + \nu_X} \frac{1}{\sin 2\eta \sin 2\psi_0} \left( \frac{\partial \epsilon_{\alpha1}}{\partial \cos \alpha} \right) \quad (5) $$

This equation is the fundamental formula of the $\cos \alpha$ method to determine the normal stress. It is rewritten as

$$ \sigma_x = K_1 \cdot M_1 \quad [\text{MPa}], \quad (6) $$

$$ K_1 = -\frac{E_X}{1 + \nu_X} \frac{1}{\sin 2\eta_0 \sin 2\psi_0} \quad [\text{MPa}], \quad (7) $$

$$ M_1 = \frac{\partial \epsilon_{\alpha1}}{\partial \cos \alpha}, \quad (8) $$

where $K_1$ is named the stress constant and $M_1$ is the slope of the $\epsilon_{\alpha1}-\cos \alpha$ plot. In Eq. (5), the variation of $\eta$ with $\alpha$ is small, so $\eta$ in Eq. (5) for $K_1$ is replaced by the complementary angle of the stress-free diffraction angle; $\eta_0 = 90 - \theta_0$ [deg].

### 2.3. Materials and specimens

Target materials are ferritic steels with small grain size. Before X-ray stress measurement, it is necessary to make sure that the entire D-S ring from specimens is continuous and of sufficient intensity. When this condition is satisfied, we are assured to have a sufficient number of diffracting grains in the irradiated area. Materials with a strong texture should be avoided.

The surface to be measured should be smooth and free from any inappropriate layer such as oxide layer. If there is severe plastic flow in the surface layer due to cutting or grinding, the out-of-plane shear stress may exist. In such cases, the assumption of the plane stress state is violated. Although the present standard does not include the details of the analysis, the triaxial
stress analysis of out-of-plane stresses is possible by the cosα method [9] and is recognized as much simpler analysis in comparison with the counterpart based on the sin²ψ method.

## 2.4. Characteristic X-ray and stress analyzer

The 211 diffraction of α-Fe by Cr-Kα radiation is adopted for stress measurements. The diffraction angle for stress-free 2θ₀ is 156.4°. Figure 3 illustrates constituent elements of a stress analyzer. It consists of an X-ray generator, a two-dimensional detector, a single exposure unit with a collimator for the incident angle ψ₀, a reader, a computer, and a controller of the detector. The goniometer system is unnecessary. Before measurements, the analyzer should be set up in the right alignment and position. To ensure the alignment, it is recommended to measure the zero stress of stress-free powder.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Stress analyzer for the cosα method.</figcaption></figure>

## 2.5. Correction of diffraction intensity

The measured profile, Φ<sub>M</sub>(ψ₀, 2θ, α), of diffraction intensity obtained from the D-S ring must be corrected by the modified Lorentz-polarization factor, LP(2θ), the absorption factor, A(ψ₀, 2θ, α), and the geometrical factor, GF(2θ). The corrected profile, Φ(ψ₀, 2θ, α), of diffraction intensity at the rotation angle α is given by

$$ \Phi(\psi_0, 2\theta, \alpha) = \frac{\Phi_M(\psi_0, 2\theta, \alpha)}{LP(2\theta) \cdot A(\psi_0, 2\theta, \alpha) \cdot GF(2\theta)}, \quad (9) $$

where three functions in the denominator are as follows:

$$ LP(2\theta) = \frac{1 + \cos^2 2\theta}{\sin^2 \theta}, \quad (10) $$

$$ A(\psi_0, 2\theta, \alpha) = \mu \cdot T(\psi_0, 2\theta, \alpha), \quad (11) $$

$$ GF(2\theta) = (\cos 2\theta / \cos 2\theta_0)^2, \quad (12) $$

and μ is a linear absorption coefficient, and T(ψ₀, 2θ, α) is the X-ray effective penetration depth given by

$$ T(\psi_0, 2\theta, \alpha) = \frac{1}{\mu} \left[ \frac{1}{\cos \psi_0} + \frac{1}{\sin 2\theta \sin \psi_0 \cos \alpha - \cos 2\theta \cos \psi_0} \right]^{-1} \quad (13) $$

## 2.6. Determination of the diffraction angle

The diffraction intensity profile is rarely separated into two peaks of Kα₁ and Kα₂ because the distance L between the detector and the specimen is short, generally less than 50 mm. Therefore, the doublet of the diffraction intensity profile is treated as a single peak, and the diffraction angle 2θα is determined by the half-value-breath method of the doublet.

## 2.7. Plotting the εα₁-cosα diagram and obtaining the slope of the εα₁-cosα plot

For several rotation angles α, a set of four strains, εα, επ+α, ε-α, and επ-α, are determined, and strain parameter, εα₁, are calculated by Eq. (3). Plot εα₁ data against cosα as shown in Fig. 4.
The relationship between ε<sub>α1</sub> and cosα is approximated by the linear regression analysis by the least-squares method. The regression line must pass through the origin, because ε<sub>α1</sub> = 0 for cosα = 0. A positive slope corresponds to a compressive stress and a negative slope to a tensile stress.

<figure><img src="image_5.png" /><figcaption>Fig. 4. The ε<sub>α1</sub>-cosα plots.</figcaption></figure>

## 2.8. Calculation of stress and confidence interval

The normal stress σ<sub>x</sub> can be calculated by substituting the obtained slope M<sub>1</sub> into Eq. (6) as follows:

$$ \sigma_x = K_1 \cdot M_1 \quad [\text{MPa}]. \tag{14} $$

The confidence interval Δσ<sub>x</sub> of the normal stress σ<sub>x</sub> can be also calculated as follows:

$$ \sigma_x \pm \Delta\sigma_x = K_1(M_1 \pm \Delta M_1) \quad [\text{MPa}], \tag{15} $$

where

$$ M_1 = \frac{\sum_{i=1}^{n} (\cos \alpha)_i (\varepsilon_{\alpha 1})_i}{\sum_{i=1}^{n} (\cos \alpha)_i^2}, \tag{16} $$

$$ \Delta M_1 = t(n-1, \gamma) \cdot \sqrt{\frac{\sum_{i=1}^{n} \{ (\varepsilon_{\alpha 1})_i - M_1 (\cos \alpha)_i \}^2}{(n-1) \sum_{i=1}^{n} (\cos \alpha)_i^2}}, \tag{17} $$

and n is the number of strain parameter data ε<sub>α1</sub>, t (n − 1, γ) is the value of t-distribution having (n − 1) degree of freedom at the (1 − γ) confidence level. In the ε<sub>α1</sub>-cosα plot, the regression line, ε<sub>α1</sub> = M<sub>1</sub> · cosα, passes through the origin.

In the standard, the confidence level (1 − γ) = 0.683 and n = 20 are adopted [7, 8], so that the value of t-distribution becomes t (19, 0.317) = 1.0277.

## 2.9. X-ray elastic constant, incident angle and stress constant

The X-ray elastic constants E' = E<sub>X</sub> / (1 + ν<sub>X</sub>) for ferritic steels is independent of the incident angle ψ<sub>0</sub> and the method of X-ray stress measurement. The subcommittee conducted the round robin test using carbon steel with 0.46 mass% C and recommended the values of the incident angle and the X-ray elastic constant E' for ferritic steels, as described in Appendix. The recommendation is the X-ray elastic constant E' = 175 GPa and the incident angle ψ<sub>0</sub> = 35°.
This $E'$ value is equal to that adopted in the standard of the $\sin^2\psi$ method for ferritic steels [10, 11, 12].

### 3. Concluding remarks

The standard adopts the 211 diffraction of $\alpha$-Fe by Cr-K$\alpha$ radiation for stress measurements of ferritic steels. The standard recommends the X-ray incident angle $\psi_0 = 35^\circ$ and the X-ray elastic constant $E' = 175$ GPa.

The following four-step protocol is described in the standard, together with necessary precautions to carry out stress measurements.

(1) First step: To obtain the 211 diffraction of $\alpha$-Fe, the entire D-S ring should be recorded by a two-dimensional detector at $\psi_0 = 35^\circ$.

(2) Second step: Strain parameters $\epsilon_{a1}$ are calculated at any rotation angle $\alpha$ from the recorded D-S ring, and then these data are plotted on the $\epsilon_{a1}$-cos$\alpha$ diagram.

(3) Third step: The strain parameter $\epsilon_{a1}$ is approximated as a linear function of cos$\alpha$ passing through the origin, and the slope $M_1$ of the regression line is evaluated by a least-squares method.

(4) Fourth step: The normal stress $\sigma_x$ is calculated for multiplying the slope $M_1$ by the stress constant $K_1$. The confidence interval $\Delta\sigma_x$ is calculated using Eq. (15).

The Japanese standard of the cos$\alpha$ method was published in 2020, and an English version is scheduled to be published in 2022.

### 4. Acknowledgements

The standard is the outcome of the activities of the X-Ray Committee of the Society of Materials Science, Japan. We wish to thank the group members who joined the round robin test and contributed to the establishment of the standard. We referred to previous Japanese standards for X-ray stress measurements by the $\sin^2\psi$ method for ferritic steels [11] and ceramics [13], and the English version of the standard [12].

## Appendix: Round robin test for standardization

Thirteen subcommittee members from academia and industry participated in the round robin test [7,8]. The round robin test was carried out to determine the X-ray elastic constant, optimal incident angle, and other experimental protocol for accurate stress measurements.

The material is carbon steel with 0.46 mass% C, whose Young's modulus and Poisson's ratio were 206 GPa and 0.26, respectively. A four-point bending stress is applied to a plate specimen, and the applied stress is calculated from the surface strain measured by a strain gauge. At each incremental step of the applied stress, the surface stress is measured by the cos$\alpha$ method using commercial stress analyzers, $\mu$-X360n or $\mu$-X360s (Pulstec Industrial Co. Ltd). Table 1 summarizes the conditions of stress measurements. The incident angle is set to be $\psi_0 = 35^\circ$ and $45^\circ$.

Figure 5 shows examples of the $\epsilon_{a1}$-cos$\alpha$ plots obtained under different applied bending stresses for the case of $\psi_0 = 35^\circ$. Each solid line is a result of linear regression using the least-squares method. All regression lines passing through the origin are well fitted to the experimental data. The slope $M_1$ of the plots decreases with increasing applied bending stress. Figure 6 shows the relationship between slope $M_1$ and the applied bending stress,

<table><caption>Table 1. Measurement conditions.</caption><thead><tr><th>Characteristic X-rays</th><th>Cr-Kα</th></tr></thead><tbody><tr><td>Incident angle, ψ<sub>0</sub></td><td>35° or 45°</td></tr><tr><td>Distance, L</td><td>42 mm</td></tr><tr><td>Diffraction</td><td>α-Fe, 211</td></tr><tr><td>Stress-free 2θ<sub>0</sub></td><td>156.4°</td></tr></tbody></table>

$\sigma_{app}$. The derivative $\partial M_1 / \partial \sigma_{app}$ is related to the stress constant $K_1$ and X-ray elastic constant, $E' = E_X / (1 + v_X)$, as follows:
$$ \frac{\partial M_1}{\partial \sigma_{app}} = -\frac{1+\nu_X}{E_X} \sin 2\theta_0 \sin 2\psi_0 = \frac{1}{K_1} \qquad (18) $$

The derivative $\partial M_1 / \partial \sigma_{app}$, i.e., the slope of the plot in Fig. 6, is also approximated by the least-squares method. The slope and confidence level of 68.3% are $-2.15 \times 10^{-6}$ [1/MPa] and $4.10 \times 10^{-8}$ [1/MPa], respectively. Then, the stress constant $K_1$ is determined as $-4.65 \times 10^5$ [MPa] and the X-ray elastic constant as $E' = 1.75 \times 10^5$ MPa.

<figure><img src="image_6.png" /><figcaption>Fig. 5. Examples of $\varepsilon_{a1}-\cos\alpha$ diagrams.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 6. Slope $M_1$ vs. applied stress $\sigma_{app}$.</figcaption></figure>

All data of the elastic constant obtained from the round robin test are plotted on the cumulative distribution function (CDF) for the normal distribution in Fig. 7 (a), where the solid circles or empty square marks show the experimental data and the solid and dotted lines are the fitted CDF curves. Although there is some discrepancy, the experimental results can be approximated by the normal distribution. The published results of ferritic steels using the $\sin^2\psi$ method are shown in Fig. 7 (b), where more than forty data are collected by the round robin test [10]. The empty diamond data agree well with the fitted CDF curve. The mean and standard deviation of $E'$ determined from the fitted CDF curves are listed in Table 2. Based on the results shown in Fig. 7 (b), the elastic constant, $E'$, of the 211 diffraction of $\alpha$-Fe by Cr-K$\alpha$ radiation is recommended as 175 GPa in the standard of the $\sin^2\psi$ method [10].

<figure><img src="image_8.png" /><figcaption>Fig. 7. Elastic constant data plotted on the CDF for normal distribution. (a) $\cos\alpha$ method; (b) $\sin^2\psi$ method.</figcaption></figure>

<table><caption>Table 2. Mean and standard deviation of X-ray elastic constant of the 211 diffraction of $\alpha$-Fe.</caption><thead><tr><th rowspan="2">Method</th><th rowspan="2">Incident<br>angle [°]</th><th colspan="2">X-ray elastic constant [GPa]</th></tr><tr><th>Mean value E'<sub>mean</sub></th><th>Standard deviation s</th></tr></thead><tbody><tr><td rowspan="2">cos&alpha; method</td><td>35</td><td>176.4</td><td>7.6</td></tr><tr><td>45</td><td>180.8</td><td>9.6</td></tr><tr><td>sin<sup>2</sup>&psi; method</td><td>&mdash;</td><td>175.4</td><td>14.6</td></tr></tbody></table>
Substituting $E' = 175$ GPa and $2\theta_0 = 156.4^\circ$ into Eq. (7) gives the $K_1$ term as follows:

$$K_1 = -\frac{4.37 \times 10^5}{\sin 2\psi_0} \text{[MPa].} \qquad (19)$$

Figure 8 shows the $K_1$ values calculated by substituting the incident angles into Eq. (19). The predicted $K_1$ values have a maximum at $\psi_0 = 45^\circ$, thus the closer the incident angle to $45^\circ$, the higher the sensitivity of the X-ray stress measurement. In the figure, the mean $K_1$ values from the round robin test results of the $\cos\alpha$ method were also plotted with the solid circle and the empty square. The solid circle is closer to its respective predicted $K_1$ value. Therefore, $\psi_0 = 35^\circ$ is found to be more appropriate to the incident angle than $\psi_0 = 45^\circ$. Furthermore, the smaller incident angle $\psi_0 = 35^\circ$ has a practical advantage because the diffraction cone is less obstructed by the sample.

<figure><img src="image_9.png" /><figcaption>Fig. 8. Predicted $K_1$ calculated by Eq. (19) and measured $K_1$ obtained by the round-robin test.</figcaption></figure>

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