**Table of Contents**show

Residual stress is the engineering name of the first type of internal stress.

The distribution of residual stress in the workpiece is generally uneven, and the residual stress will have a significant impact on the static strength, fatigue strength, shape and size stability and corrosion resistance of the workpiece.

Therefore, the measurement of residual stress is very important.

Residual stress measurement methods can be divided into destructive testing method and nondestructive testing method.

The destructive testing method is to remove part of the measured workpiece by machining, release the local residual stress and generate the corresponding strain and displacement, and calculate the residual stress of the workpiece according to the relevant mechanical principles.

The commonly used destructive detection methods are drilling method and ring core method.

The nondestructive testing method is to calculate the residual stress by establishing the relationship between the residual stress and the physical quantity that can cause the change of a certain physical quantity in the material (such as the crystal plane spacing, the propagation rate of ultrasonic wave in the material or the magnetic permeability).

Commonly used nondestructive testing methods include X-ray diffraction method, neutron diffraction method, magnetic method and ultrasonic method.

Among them, X-ray diffraction method is the most widely used method at home and abroad due to its mature principle and perfect method, and its testing equipment is becoming more and more perfect, including laboratory instruments with complete functions, portable instruments suitable for field measurement, and special testing devices suitable for special occasions.

The method of measuring residual stress by X-ray diffraction was first proposed by Russian scholar Akchenov in 1929, which equated the macroscopic strain of materials with the lattice strain.

In 1961, German scholar Macherauch developed sin^{2}ψ method based on this idea, which makes the measurement of residual stress by X-ray diffraction become a mature and operable testing technology.

After 60 years of development, the residual stress measurement technology by X-ray diffraction has developed many different measurement methods.

At present, there are two main methods for measuring residual stress by X-ray diffraction: sin^{2}ψ method and cosα method.

**1. Classification of X-ray diffraction residual stress measurement methods**

In order to master the residual stress measurement technology of X-ray diffraction, it is necessary to summarize its methods.

(1) X-ray diffraction residual stress measurement methods can be divided into sin^{2}ψ method and cosα method.

(2) sin2ψ method can be classified into 2θ method, d value method and strain method according to the calculation method of residual stress.

(3) According to the geometric relationship between ψ and 2θ, method sin^{2}ψ can be divided into two types: co tilt method and roll method.

(4) According to the scanning mode of X-ray tube and counter tube, it can be divided into fixed ψ0 method and fixed ψmethod.

(5) The roll method can also be divided into standard roll method, modified roll method and roll fixation method ψ.

(6) The positive and negative ψ measurement method is used to determine the shear stress τφ.

(7) The X-ray diffraction method is generally used to measure the stress in the specified direction at the specified point, and there are also principal stress measurement methods at the specified point.

(8) The swing method can be divided into ψ0 swing method, ψ swing method, Debye ring swing method, φ angle swing method and X/Y reciprocating translation method.

(9) In terms of diffraction geometry, there are focusing method, quasi focusing method and parallel beam method.

**2. ****sin**^{2}**ψ**** method for determination of residual stress by X-ray diffraction**

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The stress is measured by strain. For polycrystalline materials, the strain corresponding to the residual stress is considered to be the statistical result of the lattice strain in the corresponding area.

Therefore, the residual stress can be calculated by measuring the lattice strain according to the X-ray diffraction principle.

The residual stress of the material corresponds to the macro strain.

The macro strain is considered to be equivalent to the lattice strain.

The lattice strain is the relative change of the crystal plane spacing, which can be calculated by the diffraction device according to the Bragg law.

This is the complete idea of the X-ray diffraction residual stress measurement method.

**2.1 Bragg’s law**

When a beam of X-ray with a certain wavelength λ shines on the polycrystal, the maximum reflected X-ray intensity (i.e. diffraction peak) will be received at a certain diffraction angle 2θ, which is called X-ray diffraction, as shown in Fig. 1.

The X-ray wavelength λ, diffraction crystal plane spacing d and Bragg angle θ meet the formula (1).

In the X-ray diffraction residual stress analysis, the X-ray tube with the appropriate target material is selected, that is, the appropriate wavelength λ is selected, and the diffraction angle 2θ is measured by the diffraction device, then the crystal plane spacing d of the corresponding crystal plane can be calculated.

Fig. 1 X-ray diffraction geometry

**2.2 Azimuth angle of crystal plane diffraction ψ**

According to the reflection law of optics, the normal of the crystal plane participating in diffraction must be on the angular bisector of the incoming ray and the reflected ray, as shown in Fig. 2.

The included angle between the normal of the diffraction crystal plane and the normal of the sample surface is the azimuth angle of the normal of the diffraction crystal plane, which is usually expressed in ψ.

Fig. 2 Schematic Diagram of X-ray Diffraction Crystal Plane Azimuth ψ

According to Bragg’s law, the spacing d_{ψ} of crystal planes in the direction corresponding to designated ψ can be determined.

If the crystal plane spacing d_{0} in the stress-free state is known, the lattice strain ε_{ψ} in the specified orientation can be measured.

**2.3 Scope of application ****of ****sin**^{2}**ψ**** metho****d**

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S_{1}, S_{2 }and S_{3} are the coordinate axes of the specimen surface, and S_{1} is defined by the researcher.

Fig. 3 shows the coordinate system for residual stress measurement by X-ray diffraction.

Fig. 3 Coordinate system for X-ray diffraction stress measurement

According to the generalized Hooke’s law, the strains of these crystal planes are determined by the stress tensor at point O, and are closely related to the sine and cosine of φ and ψ, the Young’s modulus of the material and Poisson’s ratio.

Therefore, it is possible to obtain the three-dimensional stress of point O, including stress σ_{φ}.

The expression of strain in OP direction can be derived from elasticity.

For most materials and parts, the penetration depth of X-ray is only a few microns to tens of microns, so it is generally assumed that σ_{33}=0.

Therefore, the strain in OP direction is shown in equation (2).

The sin^{2}ψ-method formula is derived based on bragg’s law and elastic theory.

The objects involved in elastic theory are assumed to be homogeneous, continuous and isotropic media.

For polycrystalline metal materials, this assumption can be approximately satisfied only if the grains are fine and there is no texture.

Fig. 4 shows the functional relationship curve of ε_{n} and sin^{2}ψ for isotropic materials, materials with stress gradient or composition gradient, materials with shear stress, and anisotropic materials with texture.

As shown in Fig. 4c), if the shear stress τ_{13}≠ 0, τ_{23}≠0 and sin^{2}ψ curve have ± ψ bifurcation, σ_{φ} and τ_{φ} can be obtained by using the measured strain data ε_{﹢ψ} and ε_{﹣ψ} on a series of ± ψ angles, as shown in Formula (4) and Formula (5).

It should be noted that it is rare for the sin2ψ curve shown in Fig. 4c) to have a ± ψ bifurcation.

Because the penetration ability of X-ray used for diffraction to the tested material is extremely low, mostly in the depth of several microns or more than ten microns.

Therefore, the stress components perpendicular to the material surface can be considered as zero.

Only when the principal stress plane deviates from the surface of the specimen under the condition of special machining (such as powerful and large cutting amount grinding) can τ_{13}≠0 and τ_{23}≠0 occur.

The ± ψ bifurcation usually occurs, and the fitting curve often does not have the ellipse attribute, which is essentially caused by the systematic error of the ± ψ mechanism of the goniometer, so there is no need to overemphasize the necessity of ellipse fitting.

To sum up, the practical and operable process of determining the residual stress by X-ray diffraction is to select several ψ angles (or several pairs of ± ψ angles) to measure the diffraction angle 2θ_{φψ} respectively, and then calculate.

Scholars have developed many methods on how to arrange the spatial geometric relationship between ψplane and 2θ plane, how to obtain diffraction curve, how to calculate, etc.

**3. True strain method, 2 θMethod and d-value method**

The diffraction angle 2θ_{φψ} is measured by the X-ray diffraction device, and the corresponding crystal plane spacing is calculated as d_{φψ} according to the Bragg law, then the lattice strain ε_{φψ} can be expressed by the crystal plane spacing, as shown in Formula (6).

The true strain is directly substituted into Equation (3), Equation (4) and Equation (5) to calculate the stress, which is the true strain method expression.

The true strain method is adopted, and the exact values of d_{0} and θ_{0} are not required.

In most cases, the true strain method has significant advantages.

Approximate equations can also be used to calculate the strain, as shown in Eq. (7) and Eq. (8).

The calculation formula of 2θ method is shown in equation (9).

Where, K is the stress constant, and its calculation formula is shown in equation (10).

Where: ν is the Poisson’s ratio of the material.

For some materials, θ_{0} varies greatly with the chemical composition, and the results will have large deviation if the stress constant is used.

The true strain method has been included in the European Union standard EN 15305-2008 Non destructive testing — Test method for residual stress analysis by X-ray diffraction and GB/T 7704-2017 Non destructive Testing X-ray Stress Measurement Method.

XL-640 domestic stress meter lists the true strain method as the default stress calculation method, and 2θ method can be selected for calculation.

**4. Co tilt method and roll method**

The coplanar method is a measurement method in which the 2θ plane coincides with the ψplane (stress direction plane), as shown in Fig. 5.

Fig. 5 Geometric Diagram of the Same Tilt Method

With the same inclination method, the X-ray incidence angle ψ_{0} is dominant, while the angle ψ can be calculated, as shown in Eq. (11) and Eq. (12).

In the stress test of the actual workpiece, when the test point is located in a similar shallow groove, the testing space of the goniometer is limited, so the same inclination method is more suitable.

The roll method is a measurement method in which the 2θ plane and the ψ plane (stress direction plane) are perpendicular to each other, as shown in Fig. 6.

Fig. 6 Geometric Diagram of Roll Method

The characteristic of the roll method (χ method) is that the absorption factor of the diffraction peak is very small, which is conducive to improving the measurement accuracy.

The 2θ range and ψrange can be fully expanded as required. For some materials, diffractive rays with low peak positions (such as peak positions below 145 °) can be used to measure the stress.

However, since the 2θ plane and the ψ plane of this method are perpendicular to each other, what is needed is a three-dimensional space, which is difficult to apply to the measurement of some narrow parts of the space.

The stress instrument product of a foreign company adopts the modified roll method with double detectors. Its geometric layout is shown in Fig. 7.

Fig. 7 Geometric Diagram of Modified Roll Method

As early as January 1977, Li Jiabao, Institute of Metals, Chinese Academy of Sciences, proposed this test method and calculation formula, as shown in Eq. (13) and Eq. (14).

The roll method can be divided into fixed ψ_{0} method and fixed ψ method.

Fixed ψ method is superior to the former because of its accurate principle and good practical effect.

The combination of the two methods, namely, the fixed ψ method under the condition of roll, will make the absorption factor equal to 1.

That is to say, no matter whether the diffraction peak is diffuse or not, its back bottom will not tilt, the peak shape is basically symmetrical, and the peak shape and intensity will not change with the change of angle ψ in the case of no texture.

Obviously, this feature is very beneficial to improve measurement accuracy, and the roll fixed ψ method is an ideal measurement method.

**5. Swing method**

The swing method is to use each set angle ψ (or ψ 0 angle) as the center, so that the X-ray tube and the detector swing left and right at a certain angle (± Δψ Or ± Δψ 0).

This method increases the number of grains participating in diffraction, which is an approximate method to solve the problem of stress measurement of coarse grained materials.

Based on this idea, the φangle swing method and X/Y translation swing method can also be adopted, and even different swing methods can be combined for testing.

**6. Determination of residual stress by X-ray diffraction ****cosα**** method**

In 2012, PULSTEC Company of Japan launched a stress instrument based on two-dimensional detector technology for the first time.

The instrument adopts a single incident mode and uses a two-dimensional detector to collect X-ray diffraction, which can collect Debye ring information at the test point in a short time.

The angle ψ formed by the crystal face normal corresponding to each point on the Debye ring and the sample surface normal is not in the same plane, so the stress cannot be calculated by the sin^{2}ψmethod, so the angle α is used, which is the so-called cosα method, as shown in Fig. 8.

Fig. 8 Geometric Diagram of Method cosα

This test method is more suitable for the surface stress test of large steel structures.

For testing coarse grain materials or materials with texture, the use of this instrument has limitations.

The cosα method is based on the principle of elasticity, as shown in Eq. (15) and Eq. (16).

Fig. 9 The maximum acquisition range of angle ψ of “full two position detector” (the incidence angle is 45 °) is shown in Fig. 8.

Angle α is on the plane of Debye ring, that is, the center angle of each point on Debye ring.

Fig. 9 Location of data points of method sin^{2}ψ in s curve

In fact, the two methods for measuring residual stress by X-ray diffraction are based on the same mechanical principle.

The strain tensor is transformed in the space angle, and the α angle used in the cosα method can be completely converted to the ψ angle.

The cosα method is actually the sin^{2}ψ method of approximate treatment.

**7. Comparison of the residual stress of hot-rolled steel plate measured by different instruments**

Generally, the hot-rolled steel plate is considered to have no texture. In fact, some parts of the steel plate will have some texture due to various factors.

In this case, most users still prefer to use X-ray diffraction to measure the residual stress.

Select a hot rolled steel plate with texture. See Table 1 and Table 2 for test conditions and results. The test report of residual stress at point Z (0) measured by each instrument is shown in Figure 10-13.

Table 1 Test Parameters for Measuring Residual Stress of Hot Rolled Steel Plate with Different Stress Instruments

Equipment Type | μ-X360S | PROTO LXRD | X-RAYBOT | XL-640 |

test method | cosα | sin^{2}ψ | sin^{2}ψ | sin^{2}ψ |

Voltage/kV | 20 | 30 | 20 | 25 |

Current/mA | 1 | 25 | 1 | 6 |

Illuminated spot/mm | 1 | 1 | 1 | 1 |

ψ Range/（°） | – | -35~35 | -40~40 | 0~45 |

Strain calculation method | – | D value method | Strain method | Strain method |

Peak determination method | – | PersonVII | Middle point | Cross correlation method |

Residual stress/MPa | 78 | 213.6 | 144 | 113 |

Table 2 Residual Stress of Hot Rolled Steel Plate Measured by Different Stress Instruments

Test point | μ-X360S | PROTO LXRD | X-RAYBOT | XL-640 |

Z(5) | 29,47 | 122 | 107 | 77 |

Z(4) | 37,52 | 135 | 112 | 70 |

Z(3) | 74,70 | 104 | 95 | 67 |

Z(2) | 38,28 | 153 | 99 | 134 |

Z(1) | 37,64 | 166 | 122 | 101 |

Z(0) | 64,78 | 144 | 213 | 113 |

Z(-1) | 72,71 | 138 | 97 | 139 |

Z(-2) | 62,52 | 134 | 83 | 145 |

Z(-3) | 75,70 | 120 | 93 | 153 |

Z(-4) | 63,56 | 114 | 80 | 148 |

Z(-5) | 79,27 | 94 | 93 | 152 |

Fig. 10 Debye ring at point Z (0) measured with μ-X360S-type stress meter

Fig. 11 2θ-sin^{2}ψ curve of Z (0) measured by PROTO LXRD stress meter

Fig. 12 2θ-sin^{2}ψ Curve of Measuring Point Z (0) with X-RAYBOT Stress Meter

Fig. 13 ε-sin^{2}ψ-curve of Z (0) measured with XL-640 stress meter

The residual stress measured by c method is smaller than that measured by s method.

For the Z (0) test point, the stress meter is used. According to the principle of equal spacing of sin^{2}ψ, 8 ψ angles are selected within the range of 0 °~45 °.

The results are shown in Fig. 14-15. It can be seen that the sin^{2}ψ curve of the material presents a “shock” type due to the texture.

The ordinate of the sin^{2}ψ curve in Fig. 13 is strain ε. After changing the ordinate to 2θ, perform linear fitting. The results are shown in Fig. 14.

The slope M of the fitting line is -0.355, and the residual stress σis 113MPa.

Fig. 14 Fitting results of 2θ-sin^{2}ψ curve measured by XL-640 stress meter at point Z (0)

The ψ range selected by the μ-X360S stress meter is equivalent to the first two 2θ values of shielding, and then the straight line fitting is performed. The results are shown in Fig. 15.

Fig. 15 Fitting Results of 2θ-sin^{2}ψ Curve of Z (0) Measured by μ-X360S-type Stress Instrument

Use PROTO LXRD stress meter to test the selected ψ range, shield the last three 2θ values in Fig. 14, and then perform linear fitting. The results are shown in Fig.16.

Fig. 16 Fitting results of 2θ-sin^{2}ψ curve measured by PROTO LXRD stress meter at point Z (0)

It can be seen from Figure 12 that the maximum sin^{2}ψ value of point Z (0) is 0.4 by using the X-RAYBOT stress meter.

According to the selected ψ range, shield the last two 2θ values in Fig. 14, and then perform linear fitting. The results are shown in Fig. 17.

Fig. 17 Fitting Results of 2θ-sin^{2}ψ Curve of Z (0) Measured by X-RAYBOT Stress Tester

Due to the texture of the material, its sin^{2}ψ curve is oscillatory.

The selected ψ angle range is different, and the slope and residual stress of the fitting line obtained are obviously different.

In the case of unknown material texture and coarse grain, it is not allowed to select a small ψ range and a small number of ψ stations for residual stress measurement, otherwise, large measurement errors will be caused.

Linear fitting may not be reasonable for textured materials whose sin^{2}ψ curves are oscillatory.

In the actual measurement process, people usually use linear fitting to deal with the fluctuations caused by this vibration and measurement errors.

As for the range of ψ, it may not be reasonable to reach 45 ° at most. If the influence of penetration depth can be ignored, a larger angle of ψ will be more conducive to obtaining more correct results.

For coarse grain materials or materials with texture, the setting range of ψangle shall be expanded as far as possible, and the influence of nonlinear distribution of ε-sin^{2}ψ can be eliminated by measuring ± ±ψ angle.

For fitting regression straight lines with the least squares method, if the distance between independent variables is larger (the ψ range is larger) and the measured data is more (the number of ψ stations is more), the accuracy of the fitted straight lines is higher and the tested values are more reliable.

The measurement accuracy can also be improved by increasing the exposure area of X ray, or by increasing the number of participating diffraction grains by using the swing method.

**8. Conclusion**

(1) The sin^{2}ψmethod can be used to determine the residual stress by selecting a larger ψ range and more ψ stations, so as to improve the test accuracy.

Method c adopts single exposure. If the range ψis not enough, large measurement errors will be caused, which has limitations and needs to be further improved.

(2) In the measurement methods based on the principle of sin^{2}ψmethod, the roll method has obvious advantages compared with the same tilt method.

If the space condition of the measured point allows, the roll method shall be used as far as possible.

For the measurement of residual stress in the groove of some parts, the method of co-inclination is usually used.

(3) In the calculation of residual stress, the true strain method can be preferred.

(4) sin^{2}ψ method as a standard method.

The setting of angle ψ should preferably adopt the sin^{2}ψ-value bisection method, and select as many ψ angles as possible for measurement.