in Vol. 21 - March Issue - Year 2020
Accuracy in Residual Stress Measurement Methods

Figure 1a, 1b. A coil spring undergoing XRD residual stress testing on a Proto LXRD

Figure 2. A metal sample (with strain gauge rosette) before hole drilling

Figure 3. A graphic representation of diffraction

Figure 4. Residual stress measurement of a turbine blade with a portable Proto iXRD

(2), (3)

The accurate measurement of residual stress can be both challenging and time consuming, and numerous approaches have been attempted over the years with varying levels of success. Knowledge of the residual stress state of stress-critical components is extremely important in ensuring adequate performance, safety, and quality. This article will compare the accuracy of several methods of residual stress measurement.
In many of the less effective methods, multiple parameters are measured, with only one of those being the parameter of interest; therefore, it can be difficult to decouple the results. The most effective methods of residual stress measurement provide direct, quantitative data that can be used to lengthen the life of components. As will be shown below, XRD is a reliable and commonly used method, as it provides clear and accurate residual stress data at and near the surface of a component. This information can be used to pinpoint and solve stress-related problems like cracking and fatigue.

Methods of Residual Stress Measurement

Ultrasonic: The velocity of sound in a material changes in response to stress [1,2]. The magnitude of this change is a function of microstructure (i.e., hardness, crystallographic texture, dislocation density, composition, and phases present). The characteristic microstructure in a component may vary due to processes applied during fabrication, such as heat treating or shot peening; thus, it is important that calibrations are performed on the exact same material that is to be measured. This is difficult because it is unlikely to duplicate any material with identical microstructure. The depth measured by ultrasonic waves depends on the frequency used and is on the order of a few millimeters. The measured stress is an average through the thickness sampled by the acoustic wave. Ultimately, this method is qualitative due to the difficulty in decoupling residual stress from the other parameters being measured, and it requires calibration with a quantitative method.
Barkhausen: When a magnetic field is applied to a ferromagnetic sample, a noise-like signal is generated as a result of domain walls moving past defects in the crystal lattice. A coil can be used to monitor the intensity of the signal [3]. When tensile stresses are present, the intensity of the noise increases; when compressive stresses are present, the intensity decreases. The microstructure of a material has a very large influence on Barkhausen noise; therefore, as with the ultrasonic method, results of Barkhausen residual stress measurements are qualitative. In addition, this method can only be applied to ferromagnetic materials.
Eddy Current: In this method, the change in impedance (or conductivity) of a material with stress is detected by a sensor coil; this signal is then converted into a voltage [4]. The number of materials to which this method can be applied is limited, since measurement results are extremely sensitive to chemical composition, microstructure, cold work, hardness, surface roughness, and temperature [5,6]. The signal changes resulting from residual stress variations are often very small compared to those resulting from microstructure changes, making them nearly imperceptible.
The three methods discussed above have significant limitations, rendering their data inadequate for improving the quality or safety of critical parts. The methods discussed below typically provide more useful, quantitative residual stress data.
Sectioning: As a component is cut into pieces, the residual stresses equilibrate. This relaxation can be monitored by using electrical resistance or other types of strain gauges attached to the component or by physically measuring the resultant distortion. Sectioning is normally used to provide a bulk residual stress response. For complicated geometries, the technique requires finite element analysis to estimate the effect of the shape of the component [5]. The contour method, a newer sectioning method, looks at the deformation of a part’s planar surface after cutting to infer the original residual stresses normal to the cutting plane. A mathematical model is required. Some disadvantages of sectioning are its destructive nature and poor sensitivity at near-surface and surface locations.
Hole Drilling: This method involves removing stressed material by drilling a hole in the middle of a strain gauge rosette attached to the surface of the sample. The hole is typically 1-3 mm in diameter, and the measurement depth is generally limited to half of that diameter. As the hole is produced, the strain gauge detects the relaxation of the material surrounding the hole. The residual stress present in the material before drilling the hole can be calculated from the measured relaxed strains. The calculations can average over the entire depth of the hole drilled or average over each depth increment drilled [7]. This method is accurate for materials that behave linear-elastically; however, it is destructive and cannot correctly quantify stress when it is higher than 80% of the yield stress, as the drilling of the hole may cause local plastification [8].
Neutron Diffraction: The distance between crystallographic planes (d-spacing) can be used as a strain gauge: when a material is in tension, the d-spacing increases, and when a material is in compression, the d-spacing decreases. The presence of residual stresses in the material produces a shift in the diffraction peak angular position that is measured by a detector [5]. The diffraction angle, 2θ, is measured experimentally, and the d-spacing is then calculated using Bragg’s law:         nλ=2dsinθ                                              (1)

Diffraction occurs when particles such as neutrons, electrons, or photons scatter off of atoms in a crystalline lattice and the difference in the path traveled through the material is an integer multiple (n) of the wavelength of the source (λ).
Neutron diffraction can only be applied to crystalline materials. Triaxial stresses can be determined accurately if d0 (the unstressed lattice spacing) is known. The penetration depth of neutrons is on the order of millimeters to centimeters, depending on the material, and all measurements are performed below the surface. While this method is effective, it is not portable or readily accessible, and it cannot measure surface stress.
X-Ray Diffraction: Like neutron diffraction, XRD uses d-spacing to calculate residual stress. However, because the penetration depth of x-rays is on the order of 5 to 30 microns, XRD can accurately quantify stress at and near the surface of a material, which is crucial in improving the fatigue life of components [2,5,9]. XRD can be portable and used in the field to determine stress. Additionally, it can be non-destructive, which allows for repeated testing and continued use of the part after testing. Unlike in neutron diffraction, d0 can easily be approximated during a measurement, introducing only a minimal error (0.1%) in the final result.
Due to its accuracy, accessibility, and ability to measure at the surface, XRD is the most effective residual stress method for understanding fatigue life, stress corrosion cracking, weld cracking, and surface enhancements. However, caution should be taken when selecting an XRD technique, as the accuracy of XRD is dependent upon the suitability of the specific technique employed. The section below will examine three XRD techniques for their accuracy and applicability.

X-Ray Diffraction Techniques

Sin2ψ Multiple Exposure Technique (MET): The MET involves measuring d-spacings in both positive and negative ψ directions with many tilts in between, thereby sampling many different orientations of the material. The stresses can be calculated with different -ψ and +ψ tilts from approximately -45 to +45°. One normal stress (σ11) and one shear stress (σ13) can be calculated. Measurement in two additional φ directions allows full characterization of the other stress tensor components (σ22, σ23, σ12; σ33 is assumed to be 0 at the surface). Therefore, this is the preferred XRD technique for residual stress measurement, explaining its prevalence.
Single/Double Exposure Technique (SET/DET): The SET/DET are variants of the MET in which only one/two ψ angles are used. These techniques are used in an attempt to perform measurements quickly. However, the SET/DET cannot determine shear stresses, detect preferred orientation, or detect grain size problems and, hence, yield incorrect results for general-purpose use.
Cos α Technique: In the cos α technique, the position of the peak at each angle α is measured around the diffraction ring (also called the Debye ring). The strains εφψ in the direction defined by φ0, ψ0, α can be expressed by an equation in which the variables are ψ0 (the inclination of the sample relative to the goniometer), φ0 (the direction of the measurement), and 2η (the angle subtended by the incident beam and diffracted beam).
If certain stresses (σ13, σ23, σ33) are assumed to be zero, the following equations can be obtained to calculate stress in the direction of interest: 

It is clear that for a fixed ψ0, σ11 and σ13 cannot be separated and solved for. Moreover, σ11 and σ12 can be measured accurately only if σ13=0 and σ23=0.
The above equations show that the cos α method is not accurate for stress measurement except in very limited cases where the shear stresses are zero. It may be possible to use the SET/DET or cos α technique if MET measurements have confirmed that the material is homogeneous and isotropic; however, they are not applicable for general-purpose use. The XRD sin2ψ multiple exposure technique is the de facto standard today due to its accuracy and ability to consistently measure residual stress and prevent stress-related failures in a wide variety of material conditions.


[1] R. N. Thurston, "Effective Elastic Coefficients for Wave Propagation in Crystals Under Stress," J. Acou. Soc. Am, vol. 37, 1965, pp. 348-356.
[2] V. Hauk, Structural and Residual Stress Analysis by Nondestructive Methods, Elsevier, Amsterdam, 1997.
[3] S. Santa-Aho et al., "Barkhausen Noise Probes and Modelling: A Review," J Nondestruct Eval, vol. 38, no. 94, 2019, pp. 1-11.
[4] F. Yu and P. B. Nagy, "Simple Analytical Approximations for Eddy Current Profiling of the Near Surface Residual Stress in Shot Peened Metals," Journal of Applied Physics, vol. 96, no. 2, 2004, pp. 1257-1266.
[5] J. Lu et al., Handbook of Measurement of Residual Stresses, SEM, Fairmont Press, Inc., 1996.
[6] M. P. Blodgett and P. B. Nagy, "Eddy Current Assessment of Near Surface Residual Stress in Shot Peened Nickel Base Superalloys," J Nondestruct Eval, vol. 23, no. 3, 2004, pp. 107-123.
[7] M. T. Flaman et al. "Determination of Residual Stress Variation with Depth by the Hole Drilling Method," Experimental Mechanics, vol. 25, no. 9, 1985, pp. 205-207.
[8] ASTM E837-13a, "Standard Test Method for Determining Residual Stresses by the Hole-Drilling Strain-Gage Method," ASTM International, 2013, www.astm.org.
[9] I. C. Noyan, J. B. Cohen, Residual Stress Measurement by Diffraction and Interpretation, Springer-Verlag, New York, 1987.

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