# Introducing a triaxial stress field for residual stress measurements using cos-alpha techniques

Mohammed Belassel, James Pineault, Michael Brauss

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Mohammed Belassel, James Pineault, Michael Brauss. Introducing a triaxial stress field for residual stress measurements using cos-alpha techniques. ICRS 11 - The 11th International Conference of Residual Stresses, SF2M; IJL, Mar 2022, Nancy, France. hal-04015059

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# INTRODUCING A TRIAXIAL STRESS FIELD FOR RESIDUAL STRESS MEASUREMENTS USING COS-ALPHA TECHNIQUES

Mohammed Belasselª, James Pineaultª, Michael Braussᵇ

## ABSTRACT

Different techniques have been developed for the measurement of residual stress using the x-ray diffraction (XRD) method, however, significant differences in the results obtained have been observed depending on which specific technique is applied on real parts. If incorrectly applied, certain techniques may produce erroneous results under certain conditions. Observed differences in the results obtained using specific techniques (e.g. sin²ψ vs. Cosα) may depend upon the stress field assumption, biaxial versus triaxial, the collection parameters selected, the measurement conditions and geometry of the test article, the material condition and microstructure, and the processes applied that imparted the residual stress field present. A clear understanding of the strengths and weaknesses associated with the various techniques is thus a requirement in assessing their applicability to the accurate measurement of residual stress on shot peened and other components using the XRD method. This paper illustrates these differences based on simulation and experiment for all stress tensor components applied to sin²ψ, two-tilt sin²ψ and single tilt cosα. It was found that only the sin²ψ results were accurate for general use, and not adversely affected by the presence of shear stresses. The two-tilt sin²ψ and the Cosα techniques produced nearly identical erroneous stress results. Moreover, it was found that both the two-tilt sin²ψ and the Cosα are equivalent in that they both suffer from the same inadequacies and thus, **cannot** be used for general purpose applications. The only reliable and sound techniques are ones that cover the full ½-space with sufficient d-spacing sampling.

**Keywords:** Residual Stress, Biaxial, Triaxial, CosAlpha, Single Exposure, sin²Psi, Stress Standard.

## 1. Introduction

Residual stress measurements are often performed at the surface and subsurface on a variety of components with different geometries. The residual stress field induced by different processes is triaxial in nature, especially below the surface.

From the standpoint of free surface boundary condition, it is assumed that the stresses normal to the surface are zero, i.e., $σ_{13}=0$. [1]. Nevertheless, the in-plane shear stresses components $τ_{13}$ and $τ_{23}$ can indeed be present due to the non-uniformity of the strain associated with certain fabrication processes, and the material's heterogeneity. In some cases, if the material is a multi-phase one, $σ_{33}$ can also be non-zero due to the penetration of x-rays [2].

Early in their development, residual stress measurement techniques using XRD were performed using only 2ψ-tilt angles (e.g. ψ = 0° and ψ = +45°). Eventually, multiple tilt techniques were introduced, however, only the normal stress component was considered in the data analyses [3]. Later on, measurements were performed using multiple ψ tilts in both positive and negative directions (e.g. MET-Sin²ψ). The negative ψ tilts were often introduced by flipping the part 180° since the goniometers were typically equipped with half rails. This newly developed measurement procedure revealed splitting in the d-spacing versus sin²ψ data curves, known as Psi-splitting, which could be easily explained by the presence of the in-plane shear

<footnote>ª Proto Manufacturing, 6150, Morton Industrial Parkway. Lasalle, Ontario, Canada.</footnote>

<footnote>ᵇ Proto Manufacturing, 12350 Universal Drive, Taylor, Michigan, USA.</footnote>

<footnote>* Corresponding author. mbelassel@protoxrd.com</footnote>
stresses $\tau_{13}$ or $\tau_{23}$ [4]. It was also found that the magnitude of these in-plane shear stresses could increase significantly when carbides were present in the material [4]. As such, a new approach to residual stress measurements was introduced which used both the $\sin^2\psi$ technique and a triaxial stress tensor, along with new instruments equipped with double arc designed for $\psi$-positive and $\psi$-negative tilts to accommodate the more stringent data collection requirements.

This new technique (and compatible instruments) led to new standards and best practices being developed and published in many countries [5,6]. Currently, both industry and research communities adhere to these standards, especially when developing models to predict or characterize the in-service fatigue life of failure critical components. Hence, reliable residual stress measurement data is critical to the success of these models.

More recently, single tilt Cosα technique instruments, known as SET-Cosα, which collects a Debye ring in one exposure, were introduced to the market. Many instruments were acquired by novice users who were often unaware of the true limitations inherent to the SET-Cosα technique [7]. Cosα techniques that employ a single tilt (i.e. SET-Cosα) assumes a biaxial stress state, hence omit the shear stress components $\tau_{13}$ and $\tau_{23}$ in the analysis. Subsequently, it is clear that the SET-Cosα technique cannot be equated to the $\sin^2\psi$ technique which assumes a triaxial stress state. As a consequence, the SET-Cosα technique has been shown to be inaccurate and unreliable on a variety of materials as shown in many published studies [8,9,10]. In this paper we shall therefore develop the Cosα technique for a triaxial stress state using measurements and theoretical simulation, then evaluate the various parameters that affect the residual stress results when assuming biaxial stress state.

## 2. Cos-Alpha techniques

As with all angular dispersive XRD residual stress measurement techniques, the Cosα technique uses the distance between crystallographic planes (i.e. the d-spacing) as a strain gauge, and as such can only be applied to crystalline, polycrystalline, and semi-crystalline materials [1,11]. When the material is in tension, the d-spacing increases and when the material is in compression, the d-spacing decreases. The d-spacing is calculated from the measured Bragg angle $2\theta$ for a known x-ray wavelength $\lambda$ using Bragg's law:

$$n\lambda = 2dsin\theta \qquad (1)$$

Subsequently, the strain $\epsilon$ is calculated from unstressed ($d_0$) and stressed ($d_{\psi\phi}$) values for each orientation defined by the angles $\phi$ and $\psi$. Hence the strain is calculated using the following relationship:

$$\epsilon_{\phi\psi} = \frac{d_{\phi\psi} - d_0}{d_0} \qquad (2)$$

The Cosα technique uses the whole Debye ring collected with a 2D detector, and determines the peak position for each angle $\alpha$ around the ring [8]. $\psi_0$ is the inclination of the sample relative to the goniometer, $\phi_0$ is the direction of the measurement, and $2\eta$ is the angle subtended by the incident and diffracted beams. The tilting angles are therefore defined as: $\psi_1 = \psi_0 + \eta$, and $\psi_2 = \psi_0 - \eta$, where $\eta = (\pi - 2\theta)/2$ (see figure 1). Strain quantities can be defined as shown in the equations 3 and 4:

$$a_1 = \frac{1}{2}[(\epsilon_{\alpha} - \epsilon_{\pi+\alpha}) + (\epsilon_{-\alpha} - \epsilon_{\pi-\alpha})] \qquad (3)$$

$$a_2 = \frac{1}{2}[(\epsilon_{\alpha} - \epsilon_{\pi+\alpha}) - (\epsilon_{-\alpha} - \epsilon_{\pi-\alpha})] \qquad (4)$$

where $\epsilon_{\alpha}$, $\epsilon_{\pi+\alpha}$, $\epsilon_{-\alpha}$, and $\epsilon_{\pi-\alpha}$ are the measured strains from the four sectors of the ring.

For $\phi_0 = 0$, the direction of the strain can be reduced to the following expression:
$$n_i = \begin{pmatrix} \cos\eta \sin\psi_0 + \sin\eta \cos\psi_0 \cos\alpha \\ \sin\eta \sin\alpha \\ \cos\eta \cos\psi_0 - \sin\eta \sin\psi_0 \cos\alpha \end{pmatrix} \qquad (5)$$

Considering a triaxial stress state, new expressions derived from $a_1$ and $a_2$ can be calculated as:

$$a_1 = -\frac{(1+v)}{E} (\sigma_{11}\sin2\psi_0 - 2\sigma_{13}\cos2\psi_0)\sin2\eta. \cos\alpha \qquad (6)$$

$$a_2 = \frac{2(1+v)}{E} (\sigma_{12}\sin\psi_0 + 2\sigma_{23}\cos\psi_0)\sin2\eta. \sin\alpha \qquad (7)$$

<figure><img src="image_1.png" /><figcaption>Figure 1: SET-Cosα technique geometry using full Debye ring.</figcaption></figure>

Since most polycrystalline materials are considered anisotropic, Young's modulus, E, and Poisson's ratio v, are replaced by $\frac{1}{2}S_2$ and $S_1$ using the following expressions [12]:

$$\frac{1+v}{E} = \frac{1}{2}S_2 \text{ and } \frac{v}{E} = S_1 \qquad (8)$$

The equations (6) and (7) show how the normal and the shear components are related ($\sigma_{11}$, $\tau_{13}$) and ($\tau_{12}$, $\tau_{23}$) and can be determined only with a minimum of two measurements.

Cosα instrument calculation software programs generally do not consider the triaxial stress state, only biaxial, therefore the terms $\tau_{13}$, $\tau_{23}$ are completely dropped from equations 6 and 7. These equations become:

$$a_1 = -\frac{(1+v)}{E} \sigma_{11} \sin2\psi_0 \sin2\eta. \cos\alpha \qquad (9)$$

$$a_2 = \frac{2(1+v)}{E} \sigma_{12} \sin\psi_0 \sin2\eta. \sin\alpha \qquad (10)$$

It is clearly evident that these new equations (9 and 10) sacrifice the reliable and accurate residual stress measurements in exchange for speed by reducing the number of measurement directions. It can therefore be shown that this technique is extremely limited, and is considered to be inferior for general use as compared to the $\sin^2\psi$ techniques that were developed as far back as the 1970s. [1,4,11]. Moreover, when comparing multiple tilt cosα (MET-Cosα) side by side with multiple tilt $\sin^2\psi$ (MET-Sin$^2\psi$), the equations (6 and 7) can lead to the same reliable and accurate results. Equation (11) illustrates the $\sin^2\psi$ equation that can either be solved for both normal and shear stresses, or for the full stress tensor when multiple φ directions are measured.

$$\varepsilon_{\varphi\psi} = \frac{1+v}{E} (\sigma_{11}\cos^2\varphi + \sigma_{22}\sin^2\varphi + \sigma_{12}\sin2\varphi)\sin^2\psi \\ - \frac{v}{E}(\sigma_{11} + \sigma_{22}) + \frac{1+v}{E}(\sigma_{13}\cos\varphi + \sigma_{23}\sin\varphi)\sin2\psi \qquad (11)$$

The normal stress in the direction of interest, φ, can be defined as:
$$ \sigma^{\varphi} = \sigma_{11} \cos^2 \varphi + \sigma_{22} \sin^2 \varphi + \sigma_{12} \sin 2\varphi \quad (12) $$

$$ \tau^{\varphi} = \sigma_{13} \cos \varphi + \sigma_{23} \sin \varphi \quad (13) $$

### 3. Two-tilts method Sin²ψ (SET-Sin²ψ)

The SET-Sin²ψ uses either a single tilt with two detectors (two ψ angles) or 2 tilts with one detector. In general, a single tilt is used with two detectors where the angles are defined as: ψ₁ = (ψ₀ + η), and ψ₂ = (ψ₀ - η), where η = (π-2θ)/2 [1]. These angles correspond to α = 0° and α = 180° on the Debye ring (see figure 1). 2η is the angle subtended by the incident and diffracted beams. For the SET-Sin²ψ,

Equation (4) can thus be reduced to the following equation for σ₂₂=σ₃₃=τ₁₃=0 and φ=0:

$$ \sigma^{\varphi} = \sigma_{11} = \frac{\partial \varepsilon_{\psi}}{\partial \sin^2 \psi} \frac{E}{1+\nu} \quad (14) $$

Given these assumptions, when using the SET-Sin²ψ, only one normal stress component (σ^φ) can be determined. This kind of measurement can be achieved by either a single tilt with two detectors or two tilts with one detector.

### 4. Residual stress simulation

In order to demonstrate the limitations of both the SET-Cosα technique and the SET-Sin²ψ technique, we opted for a theoretical approach using simulation since it is advantageous for studying a variety of parameters [10]. The simulation consists of calculating the stress values from a known initial stress tensor σᵢⱼ and the strains εᵦᵦα in every direction for each technique for all 3 techniques: SET-Cosα, SET-Sin²ψ and MET-Sin²ψ.

The simulation consists of the following steps:

1. Calculate the strain tensor using Hooke's law:

$$ \varepsilon_{ij} = S_{ijkl} \sigma_{ij} \quad (15) $$

For an isotropic stress state where Sᵢⱼₖ is the compliance tensor. The equation can be written as:

$$ \varepsilon_{ij} = \frac{1+\nu}{E} \sigma_{ij} + \frac{\nu}{E} (\sigma_{11} + \sigma_{22} + \sigma_{33}) \delta_{ij} \quad (16) $$

Where δᵢⱼ is the Kronecker delta, δᵢⱼ = 0 if i≠j and δᵢⱼ = 1 if i=j.

2. Calculate strain in the direction of interest n defined by φ, ψ, and φ₀,ψ₀, α for the sin²ψ or cosα techniques, defined in equation (3) and equation (11), respectively:

$$ \varepsilon_n = n_i n_j \varepsilon_{ij} \quad (17) $$

3. Calculate the simulated stress using the theory applied to each technique with the conditions set.

This simulation will illustrate the effect of the inclination angle ψ₀ on the normal stress, the stress error due to the presence of shear stress, the stress evolution under load and residual stress value offset due to the presence of shear stress.

### 5. Experiments

The residual stress measurements consisted of measurements under 4-point bend loading of on an AISI 4142 steel coupon fabricated with a finish ground surface in accordance with the ASTM-E1426 standard [12]. All of the measurements were recorded using Cr-Kα (λ=2.291Å) x-radiation diffracting from the {211} hkl plane at approximately 2θ=156.4°, and ½S₂=5.14.10-6 MPa⁻¹. The two-tilt SET-Sin²ψ measurements were performed at a tilt angle ψ₀=+25° corresponding to ψ₁=37° and ψ₂=13° in a two-detector SET-Sin²ψ setting.
# 6. Results

## 6.1. Residual stress results before 4-point bend loading

The residual stress tensor was measured in the coupon prior to loading using the MET-Sin²ψ technique to evaluate the initial residual stress levels.

$$ \text{The stress tensor was found to be } \sigma_{ij}(\text{MPa}) = \begin{pmatrix} -4 \pm 10 & +6 \pm 7 & -48 \pm 2 \\ +6 \pm 7 & -258 \pm 10 & -5 \pm 2 \\ -48 \pm 10 & -5 \pm 2 & 0 \pm 6 \end{pmatrix} $$

As is illustrated above, a low normal stress value of -4 MPa was generated in the direction perpendicular to the longitudinal direction of the coupon, whereas a relatively high shear stress of -48 MPa was found to be generated by the grinding process. Using the same data set with the residual stress measured in the longitudinal direction at φ=0° and using the SET-Cosα approach, with ψ₀=+25° (corresponding to ψ₁=37° and ψ₂=13°) in a two-detector setup, the stress results for both MET-Sin²ψ and SET-Cos-α were found to be:

MET-Sin²ψ: σ=-8 ± 9 MPa and τ=-42 ± 4 MPa, SET-Cosα: σ= +85 MPa.

This represents a 93 MPa difference in the normal stress measurement result when comparing MET-Sin²ψ and SET-Cosα.

## 6.2. Residual stress results under 4-point bend loading

The simulated and the experimental results are in agreement for each given individual technique (i.e. MET-Sin²ψ, SET-Sin²ψ and SET-Cosα) both before loading and under loading. However, the results for the MET-Sin²ψ are different than the SET-Sin²ψ and the SET-Cosα results (see figure 2). The slopes of the curves of total stress versus applied stress were found to be similar in value for all techniques because of the similar relative increase in the stress values for each loading increment.

<figure><img src="image_2.png" /><figcaption>Figure 2: Simulated stress under loading for the SET-Sin²ψ, SET-Cosα in black, and MET-Sin²ψ in red, σs is the total measured stress. Dashed lines are the experimental linear fits and the solid lines are the simulated data [9].</figcaption></figure>

The values of the slopes were 1.068 and 1.039 for the SET-Sin²ψ and MET-Sin²ψ, respectively, as illustrated in figure 2. The strain values measured using the strain gauge are not
corrected for misalignment of the strain gauge orientation on the coupon and these values are estimated to be within the experimental error.

### 6.3. Effect of shear stress on the normal stress levels

The presence of shear stresses $\tau_{13}$ and $\tau_{23}$ in the material can have a significant effect on the measured normal and shear stresses $\sigma_{11}$ and $\tau_{12}$ as shown in figure 3 when using SET-Sin²$\psi$ and SET-Cos$\alpha$. The absolute stress errors in both $\sigma_{11}$ and $\tau_{12}$ are proportional to the magnitude of the shear stress only, not the magnitude of the normal stress. However, the relative error is inversely proportional to the normal residual stress magnitude level. Therefore, the relative error is higher at low stress values and lower at high stress values.

As shown in figure 2, high

shear stress magnitudes can easily be generated in ground and machined materials or components while still having relatively low magnitude normal stress values. The stress error in such cases can be significant and cannot be discounted. Moreover, the shear stress $\tau_{12}$ is strongly altered by the presence of the shear stress $\tau_{23}$ and as such can have huge implications when considering principal stress calculations.

<figure><img src="image_3.png" /><figcaption>Figure 3: Simulated stress errors caused by the presence of shear stresses $\tau_{13}$ and $\tau_{23}$ at different levels.</figcaption></figure>

### 6.4. Effect of inclination angle on simulated stress value

Residual stresses were simulated for different $\psi_0$ values for both the positive and the negative $\psi$-tilt sides with a given set of input stress tensor values:

$$ \sigma_{ij}(\text{MPa}) = \begin{pmatrix} 500 & 100 & 100 \\ 100 & 500 & 100 \\ 100 & 100 & 0 \end{pmatrix} \text{ and } \sigma_{ij}(\text{MPa}) = \begin{pmatrix} 500 & 100 & 50 \\ 100 & 500 & 50 \\ 50 & 50 & 0 \end{pmatrix} $$

The results shown in figure 4 clearly illustrate that the results for SET-Sin²$\psi$ and the SET-Cos$\alpha$ techniques are equal [8,9] in value, and the presence of the shear stress components $\tau_{13}$ and $\tau_{23}$ equally offset the stress $\sigma_{11}$. This finding was also confirmed experimentally using an SET-Cos$\alpha$ instrument [13,14]. The offset for $\sigma_{11}$ varies both with the $\psi$-tilt angle and the direction of the $\psi$-tilt, i.e., varies for equivalent positive vs. negative $\psi$-tilts. Similarly, the value of $\tau_{12}$ can vary with the $\psi$-tilt value and the direction of the $\psi$-tilt. This clearly demonstrates that the measured residual stress values can be either overestimated or underestimated depending on the orientation of the part. Both components $\sigma_{11}$ and $\tau_{12}$ converge to their true values of 500 MPa and 100 MPa when the $\psi$-tilt angles are either ±45° and ±90° respectively.

The influence of $\tau_{23}$ on $\tau_{12}$ is most significant when the stress levels are of the same order of magnitude. The effect of the tensor terms $\tau_{13}$ and $\tau_{23}$ on $\sigma_{11}$ and $\tau_{12}$ respectively are independent, however, the presence of $\tau_{23}$ will affect $\sigma_{22}$ measured in the perpendicular direction at $\phi_0=90°$.
<figure><img src="image_4.png" /><figcaption>Figure 4: Simulated stress tensor terms $\sigma_{11}$, MET-Sin$^2\psi$, SET-Sin$^2\psi$, SET-Cos$\alpha$ Methods versus $\psi_0$, for both $\psi_0<0$ and $\psi_0>0$ values using: SET-Cos$\alpha$, SET-Sin$^2\psi$, and MET-Sin$^2\psi$, for 2 different stress tensors inputs.</figcaption></figure>

## 7. Discussions

The presence of shear stresses $\tau_{13}$ and $\tau_{23}$ in materials are a common result of manufacturing processes. This is a fact that cannot be ignored as illustrated on figure 2 [11]. The inclination angle and the level of shear stresses generate stress errors when using SET-Cos$\alpha$ and SET-Sin$^2\psi$. This is primarily originated by assuming a biaxial stress field rather than a triaxial stress field is present. The consequences of using SET-Cos$\alpha$ and SET-Sin$^2\psi$ can cause over or underestimated stress results. The errors associated with these techniques are significant and be exacerbated when measuring on curved surfaces and geometries. In addition, these errors can be higher when dealing with either heterogeneous materials, elastic and plastic anisotropy, large grain size, or preferred orientation [10,15,16]. Moreover, the presence of stress gradients cannot be corrected for because of the limited data collected at depth below the surface in one single exposure. Ultimately, the use of the SET-Cos$\alpha$ technique is a step backward in progress. It effectively reverts XRD residual stress measurement techniques back to the level of unreliability that were present prior to the discovery of the shear stress, and ignores the inherent improvements in accuracy concomitant with the advent of MET-Sin$^2\psi$ techniques in general.

As demonstrated using simulation, the triaxial stress tensor is the only viable assumption capable of providing reliable and accurate stress results. Reliable stress measurements, even in one direction, requires sampling of d-spacings using both $+\psi$ and $-\psi$ tilts to cover ½-space in the same conditions as MET-Sin$^2\psi$, i.e., SET-Cos$\alpha$ and SET-Sin$^2\psi$ covers only ¼-space.

Clearly, it can be deduced from this study that the triaxial stress tensor assumptions are a much better alternative when compared to the biaxial stress tensor assumptions which can easily produce erroneous results. The triaxial stress tensor assumptions can in principle, be easily incorporated into an MET-Cos$\alpha$ measurement approach which would require a sufficient sampling of $\psi_0$-tilts so as to be able to both solve for the $\tau_{13}$ and $\tau_{23}$ components accurately, and to cover the full ½-space. Furthermore, all of stress tensor components can be determined experimentally by using a sufficient number of $\psi_0$-tilts [17]. In this case, it will be equivalent to the MET-sin$^2\psi$ techniques that is widely implemented in most instruments, and adopted by all residual stress measurement standards and best practices that use the XRD method.

## 8. Conclusions

As demonstrated in this study, the SET-Cos$\alpha$ technique is neither viable for general-purpose triaxial residual stress measurements, nor even suitable for stress measurement when assuming a biaxial stress tensor. The reduced d-spacing data sampled in ¼-space cannot deal with the
types of microstructural properties that may be present in different materials such as grain size, anisotropy, texture, secondary/tertiary phases, or stress gradients normal to the surface.

A robust and reliable measurement solution demands adherence to a triaxial stress tensor analysis, and data collection strategies that cover the full ½-space; otherwise, the realization of reliable residual stress measurement data cannot be ensured.

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