Research on Improving the Accuracy of Welding Residual Stress of Deep-Sea Pipeline Steel by Blind Hole Method

Abstract

For a typical pipeline-lifting deep-sea mining system, marine mining pipelines have higher requirements for steel welding. After the pipeline steel is welded, a large amount of residual stress will be generated inside the component because the heat treatment is not carried out, causing the pipeline to form stress corrosion cracks in the seawater environment. For high residual stress, the blind hole method has inaccurate measurement accuracy. Based on the theory of solid mechanics, this paper comprehensively considers the distance between the strain grid and the center of the hole. It introduces the ratio γ of the applied stress to the yield strength of the specimen. The strain relief coefficients under the values were investigated. The variation trend of the strain release coefficient distribution with the γ value of the three kinds of strain gauge flowers commonly used in China was discussed. Finally, the variation law of strain release coefficient with γ value was obtained by fitting. The strain release coefficient was corrected to improve the accuracy of the blind hole method for measuring high residual stress, which provides a reference value for the engineering application of the blind hole method. This has important theoretical significance and engineering value for guiding the welding construction of high-strength pipeline steel in a deep-sea environment.

1. Introduction

With the increasing social demand for energy and minerals, the development and utilization of seafloor mineral resources has gradually become an important strategic goal of all countries [1,2,3]. The main difficulty in developing and utilizing submarine mineral resources lies in the research and development of machinery and equipment for mining and transportation of mineral resources. China’s mining system comprises four subsystems: ore collection, ore lifting, surface mining ship, and measurement and control power [4,5,6] (Figure 1). As a typical pipeline-lifting deep-sea mining system, marine mining pipeline transportation plays an essential role in the efficient and safe operation of deep-sea mining. Most of the standard submarine pipeline steel is X65 and X70 grade. To adapt to the development trend of high-pressure transportation, high wall thickness, and the deep water of long-distance submarine pipelines, the grade of marine pipeline steel will need to be higher [7].

When connecting steel structures, the most commonly used process is welding. Welding is considered one of the most effective, reliable, and economical manufacturing methods for permanently connecting metals. Welding is a local rapid heating and cooling process due to the uneven heating of pipeline steel in the construction process, resulting in differences in microstructure and properties of welded joints and making it the weakest area of the whole pipeline [8,9,10,11]. However, with the improvement of the grade of pipeline steel, it is often easy to produce residual stress in the welding process. Due to the poor service environment of the mining pipeline, stress corrosion cracking caused by welding residual stress is likely to occur around the weld during the working process. This is undoubtedly fatal to the straight mining pipe in a harsh environment, especially marine pipes. Therefore, residual stress is one of the critical factors affecting the safe service of welded structures [12,13,14]. However, residual stress is inevitable in and around the welding zone [15,16]. Therefore, when an unexpected failure occurs, it is usually due to the combination of residual stress and applied stress, which seriously reduces the reliability of welded structures [17,18,19]. Especially for the components in the marine environment, it is easy to produce stress corrosion under the combined action of static stress and seawater corrosion. Corrosion cracking is the result of the combined action of stress and corrosion. If there is only one aspect, stress or corrosion, the damage may not occur. However, when both act together, it is likely that existing cracks may propagate faster. Therefore, the required stress and corrosion may be very weak but significantly impact the components when stress corrosion occurs. Because of this, stress corrosion is often ignored, leading to the continuous occurrence of accidents. Therefore, an in-depth study of the state of residual stress is of great significance in ensuring the reliability and safety of components. It has essential theoretical meaning and engineering value to guide the welding construction of high-strength pipeline steel in a deep-sea environment.

In 1934, the German scholar J. Mathar [20] proposed using the drilling method to measure the effective prestress of members. Because of its simple operation and minor damage to members in practical application, this method has been paid more and more attention. The basic principle is to drill a round hole in the component by mechanical cutting to release the stress in its field. By measuring the strain value caused by the released stress, the working stress at the measuring point of the component can be calculated. According to the depth of drilling, the drilling method can be divided into the through-hole method and the blind hole method. The size of the optical fiber’s rigid structure is often significant. If the through-hole method is used, the construction is complex. The damage to the member is substantial; therefore, the blind hole method is usually used, which effectively reduces the damage to the member caused by effective prestress detection. G.S. Schajer [21] introduces the recent progress in the measurement and analysis of residual stress by borehole method in the field of full field of view optical measurement and inversion algorithm and points out some promising directions for future development. Xu et al. [22] obtained the best pore diameter, hole depth, and strain gauge position using the finite element software ANSYS to calculate the strain change before and after drilling. The effective stress of a prestressed concrete beam was also obtained using relevant theory. For the steel plate with a round hole in the center, according to the actual stress field and the test results of the right-angle strain rosette, Wang et al. [23] calculated the node strain values under various working conditions via elastic-plastic FEM. The change of node value simulates the data measured by the right-angle strain rosette. The relationship between residual stress and test strain in the borehole testing method is established. Based on theoretical analysis, Liu et al. [24] tested the stress release value at the edge of the hole, deduced the approximate formula for stress calculation according to the test value, and evaluated the prestress accurately and effectively. It is also proposed that the residual stress is released when the drilling depth to diameter is 1.2. Wang et al. [25] studied the factors that may cause errors in the blind hole test, such as the additional stress caused by the cutting heat and the plastic change of the hole edge. They put forward corresponding methods to reduce the error and improve the accuracy of measuring residual stress by the blind hole method. Scaramangas et al. [26] obtained a modified formula for solving the plastic deformation of residual stress according to the elastic solution based on the study of plastic strain at the edge of the hole. Maxwell et al. [27] carried out drilling tests to study the factors affecting the residual stress and concluded that the residual stress release value of the component surface does not always increase with the drilling depth, but tends to be stable after increasing to the maximum value.

In measuring high residual stress, the total strain includes plastic deformation caused by drilling, pure elastic release strain, and plastic strain caused by stress concentration. The plastic drilling strain can be measured by drilling on the non-stress test plate. The modification methods for the plastic strain of the stress concentration at the edge of the hole are as follows. The plastic modification curve of the strain release coefficient of the material is obtained through the tensile test. The plastic correction formula of the strain release coefficient based on the specific energy parameter S of hole shape change is made, and the iterative correction method and the strain release coefficient classification method are used. However, the size of the strain gauge used for residual stress measurement in China is quite different from that of other international standards. For “thin” workpieces, where a through-hole is to be used, the international standard (ASTME837-08) requires that the workpiece thickness of the strain gauge flower should not exceed 0.4 D (D is the center distance of the hole piece), while the workpiece thickness of the strain gauge used in China is about 0.78 D. At present, the strain release coefficient in China is only for low residual stress. For the high residual stress, with residual stress values greater than 1/3, the existing methods do not consider the plastic strain at the edge of the blind hole. This will lead to a problem with measurement accuracy and dramatically limits the use of the blind hole method. Based on the theory of elastoplastic mechanics and comprehensively considering the distance between the strain grid and the center of the hole, a three-dimensional finite element model is established using finite element software. The stressed environment of a blind hole in the stress release state of steel with different yield strengths is simulated. The strain release coefficient under different ratios γ of applied stress to specimen yield strength is studied. The variation law of elastic strain and plastic strain under different γ values analyzes the variation trend of the strain release coefficient with γ value. The distribution reasons for the variation trend of strain release coefficients with γ values of three strain gauges widely used in China are discussed. Finally, the variation law of the strain release coefficient with γ value is obtained by fitting, which provides a reference value for the engineering application of the blind hole method.

2. Correction of Strain Coefficient in the Blind Hole Method

2.1. The Basic Principle of Measuring Residual Stress via the Blind Hole Method

The principle of measuring residual stress via the blind hole method [28] is mainly based on the assumption that there is a certain state of residual stress field and strain field in a certain region of isotropic materials, and the maximum and minimum principal stresses are ${\sigma }_1$ and ${\sigma }_2$, respectively, as shown in Figure 2. Paste a strain rosette on the surface of the area, and then drill a small blind hole in the center of the strain rosette, where the residual stress of the metal will be released. A certain amount of released strain will be produced around the blind hole so that the stress field will reach a new equilibrium. The new stress field and strain field are formed, the release strain is measured, and the test point’s initial residual stress can be calculated using the corresponding formula. The calculation formula for measuring residual stress via the blind hole method is based on the through-hole method. For the release strain measured by the strain gauge, the stress calculation formula is as follows:

$${\sigma }_{1,2}=\frac{{\epsilon }_1+{\epsilon }_2}{4A}\mp \frac{\sqrt{{\left({\epsilon }_1-{\epsilon }_3\right)}^2+\left(2{\epsilon }_2-{\epsilon }_1-{\epsilon }_3\right)}}{4B}$$

(1)

$$\mathit{tan}2\theta =\frac{2{\epsilon }_2-{\epsilon }_1-{\epsilon }_3}{{\epsilon }_3-{\epsilon }_1}$$

(2)

In the formula, A and B are the strain release coefficients, and $\epsilon$is the strain.

The strain release coefficients A and B are constant in the elastic range. When the material at the edge of the hole yields, the value of plastic strain varies with the stress level, and A and B will also have a series of values. In the calibration test of the corresponding variable release coefficient, it is assumed that the unidirectional stress field is applied in the component (${\sigma }_1=\sigma$, ${\sigma }_2=0$), and the strain gauges R1 and R2 are parallel to${\sigma }_1$, ${\sigma }_2$direction, respectively that is$\gamma =0$. In this case, the stress calculation formula can be simplified as follows:

$${\sigma }_{1,2}=\frac{{\epsilon }_1+{\epsilon }_2}{4A}\mp \frac{{\epsilon }_1-{\epsilon }_3}{4B}$$

(3)

By substituting the stress state into the Formula (3), the following results can be obtained:

$$A=\frac{\left({\epsilon }_1+{\epsilon }_3\right)}{2\sigma }$$

(4)

$$B=\frac{\left({\epsilon }_1-{\epsilon }_3\right)}{2\sigma }$$

(5)

Therefore, the release strain is ${\epsilon }_1$, ${\epsilon }_3$. The corresponding strain release coefficients A and B can be calculated according to the form

2.2. Calibration of Strain Release Coefficient by Finite Element Method

Using ANSYS software, a three-dimensional finite element model was established to simulate the stress environment of a blind hole in the stress release state of steel with different yield strengths in the blind hole calibration test to obtain more reliable A and B values to generally describe the steel.

2.2.1. Establishment of Calibration Model of Strain Release Coefficient

The calibration test is to pre-paste the strain gauge on the specific specimen, apply a unidirectional load to the specimen, drill a blind hole at a certain depth, and record the strain change (i.e., release strain) before and after drilling with the strain gauge. The specific values of strain release coefficients A and B are calculated from the formula [29,30,31]. During the experiment, samples were taken at 1/4 of the sheet forming direction according to the standard. Through the tensile test, the mechanical properties of this batch of plates were obtained. The calibration of the variable release coefficient was simulated by ANSYS software. Taking the calibrated specimen as the prototype, a cylindrical hole with a diameter of 1.5 mm and a depth of 2 mm was set on the plate model to simulate the stress concentration in the tensile process of the specimen. During the simulation process, due to the symmetry of the specimen, 1/4 of the middle part of the calibration specimen was taken to establish an ANSYS finite element model. The mesh division is shown in Figure 3.

The elastic–plastic linear strengthening model was adopted, as shown in Figure 4. The elastic–plastic linear strengthening model is also called the bilinear strengthening model, and its specific expression is as follows.

$$\left.\begin{array}{ll}\sigma =E\epsilon & \left(\epsilon \le {\epsilon }_s\right)\\ \sigma ={\sigma }_s+E^{\prime }\left(\epsilon -{\epsilon }_s\right)& \left(\epsilon >{\epsilon }_s\right)\end{array}\right\}$$

(6)

In the formula, E is the elastic modulus, $E^{\prime }$ is the tangent modulus, and ${\sigma }_s$ is yield strength. The physical parameters of the material are shown in

Table 1. Physical parameter of materials.

Elastic Modulus E/Pa	Poisson’s Ratio µ	Density ρ/(kg·m−3)	Tangent Modulus E’/Pa
2.1 × 1011	0.29	7820	6.1 × 109

.

Using the Mise yield criterion, the whole model is a hexahedral element subjected to unidirectional uniform load F. In the simulation process, F increases gradually until it reaches the yield limit of the material. Because of the symmetry, the X-direction displacement of all nodes on the YOZ plane is 0 and the Y-direction displacement of all nodes on the XOZ plane is 0.

2.2.2. Analysis of Calculation Results

Taking the yield strength and external load F of different steels as variables, the A and B coefficients were calculated according to the $\epsilon$ values in the simulation results. Comparing the calculation results under the assumption that the tangent modulus of steel is consistent, when the ratio of applied stress to specimen yield strength is the same, the strain release coefficient of steel with different yield strengths is constant. The variation curves of strain release coefficients A and B under the different applied stress ratios to specimen yield strength γ are shown in Figure 5. The inflection point of the strain release coefficient curve is mainly when the applied stress reaches 0.4 ${\sigma }_s$, 0.7 ${\sigma }_s$, and 0.9 ${\sigma }_s.$

The recommended values of strain release factors A and B are divided into four levels in the standard method for measuring residual stress by the borehole strain method (SL499-2010). For this reason, the strain variation at the measured position of the strain gauge with the applied stress is studied in the calculation results. The elastic and plastic strain curves of the measured position of the strain gauge are shown in Figure 6. With the increase of the ratio γ of the applied stress to the yield strength of the specimen, the elastic tensile strain ${\epsilon }_{e1}$ in the applied stress direction increases gradually, and the elastic compressive strain ${\epsilon }_{e2}$ in the perpendicular direction to the applied stress increases first and decreases when the γ value is about 0.7. When the γ value reaches 0.9, the elastic pressure strain value shows an increasing trend. For the plastic strain, it can be seen from the diagram that before the γ value reaches 0.7, the plastic strain in both directions is zero. When the γ value reaches 0.7, the plastic compressive strain ${\epsilon }_{p2}$ in the vertical direction of the applied stress begins to increase. When the γ value reaches 0.9, the plastic tensile strain begins to occur in the applied stress direction, and the strain values in both directions increase rapidly.

According to the theory of elasticity, when a circular hole is opened on an infinite plate, the stress concentration factor at the edge of the hole under uniaxial tension is 3. To avoid plastic deformation at the edge of the hole and ensure that the specimen is loaded repeatedly in the elastic range, the actual stress caused by the maximum load in the working part is not greater than 0.3 $R_{eL}$ during tension. Therefore, when the external stress is more than 0.3 times the yield strength ${\sigma }_s$ under the action of concentrated stress, the stress value of the element around the circular hole begins to be greater than its yield strength, and this change increases the stress borne by the component closer to the hole. In the actual measurement, there is a certain distance between the sensitive grid of the strain gauge and the center of the measuring point, as shown in Figure 7. Therefore, when the external stress reaches 0.4 ${\sigma }_s,$ the strain release coefficients A and B calculated by strain begin to change. It can be seen from Figure 5 that when the applied stress is greater than 0.75 ${\sigma }_s$, the plastic strain begins to occur at the sensitive grid used to measure the first principal strain, and then the plastic strain increases rapidly, resulting in a change in the curve slope of strain release coefficients A and B. Similarly, when the applied stress is greater than 0.9${\sigma }_s$, the plastic strain begins to occur at the sensitive grid used to measure the third principal strain, resulting in a change in strain release coefficients A and B and the curve slope.

To make the revised results more applicable, according to the use of residual stress and strain gauges in the current market, the most commonly used strain gauges BE120-1CA and BE120-3CA of AVIC Electric Measurement Company, BX120-1CG of Huang Yan strain gauge Factory, and TJ-120-1.5-φ 1.5 of Zhengzhou Machinery Research Institute were selected as the research objects. The distance between the sensitive grid of the above four strain gauges and the center of the measuring point (D value) is shown in Figure 7.

The strain correction coefficients A and B of residual stress measured by each strain gauge are compared and studied. Because the sensitive grid of strain gauge BX120-1CG of Huang Yan strain Technology Factory and strain gauge TJ-120-1.5-φ 1.5 of Zhengzhou Machinery Research Institute is the same distance from the center of the strain gauge, the simulation results of both are the same. Here, only the strain gauge BX120-1CG of Huang Yan strain Technology Factory is analyzed. The comparison results are shown in Figure 8 and Figure 9. As can be seen from the diagram, the change rules of each strain gauge are different. There is a particular gap between the sensor grid and the center of the strain gauge for different types of strain gauges.

In the comparison process, we first observed that the distance between the sensor grid and the center of the strain gauge BE120-1CA, BE120-3CA, and BX120-1CG of Huang Yan strain gauge factory was 3 mm, 5.13 mm, and 2.57 mm, respectively. Due to the stress concentration at the edge of the hole, the stress value around the blind hole varies with the distance from the blind hole, as shown in Figure 10. The specific manifestations are as follows:

(1)

On the y-axis ($\phi =\frac{\pi }{2}$), the circumferential normal stress is ${\sigma }_{\phi }=q\left(1+\frac{1}{2}\frac{r^2}{{\rho }^2}+\frac{3}{2}\frac{r^4}{{\rho }^4}\right)$. On the y-axis, the circumferential normal stress reaches the maximum value of 3q at the hole’s edge and approaches q sharply, as it is far away from the hole’s edge.

(2)

On the x-axis ($\phi =0$), the circumferential normal stress is ${\sigma }_{\phi }=-\frac{q}{2}\frac{r^2}{{\rho }^2}\left(3\frac{r^2}{{\rho }^2}-1\right)$. On the x-axis, the circumferential normal stress reaches the minimum value −q at the edge of the hole and becomes 0 at $\sqrt{3}$r, that is, the stress changes sign at this distance and becomes compressive stress. Then, as it is far away from the hole’s edge, it becomes tensile stress and gradually trends toward 0.

It can be seen from Figure 10 that the stress concentration near the orifice has two characteristics: (1) The stress near the hole is highly concentrated and (2) concerning the locality of stress concentration, the range of stress disturbance is mainly concentrated in the range of 1.5 d (d is the aperture) from the hole’s edge. Outside this range, it can be ignored. Therefore, the strain at the sensitive gate is affected by its position, which leads to a change in the measurement results of the strain gauge. Thus, the different changing laws of strain relief coefficient curves are mainly due to external loads and concentrated stress effects.

2.2.3. Correction of Strain Release Coefficient

According to the calculation results, the polynomial formula is used to fit the relationship between the variable release coefficient A and B coefficient and the ratio γ of the initial value of residual stress to the yield strength of the specimen, and the R2 values of the appropriate formulas are all greater than 0.9. This shows that the proper condition is good. The variation law of strain release coefficient with γ value is obtained based on the calculated results. The applicable requirements and specific values of strain release coefficients A and B are shown in

Table 2. Applicable conditions and specific values of strain release coefficients A and B.

Resistive Grid Base			Aperture D0 (mm)	Hole Depth h (mm)	h/D0	Hole Center Distance (mm)	The Ratio γ (σc/σs)	Strain Release Coefficient
Strain Gauge Type	Length × Width (mm)	Resistance Value (Ω)						A (×10−7 mm2/N)	B (×10−7 mm2/N)	Drilling Strain Correction $\left({\mu }_c\right)$
BX120-1CG	1 × 1	120	1.5	2.0	1.33	2.57	0 ≤ γ ≤ 1	−946.04γ6 + 3180.2γ5 − 4025.5γ4 + 2465.2γ3 − 750.71γ2 + 104.87γ − 18.829	−1211.4γ6 + 3171.7γ5 − 3358.4γ4 + 1818.2γ3 − 513.54γ2 + 68.932γ − 40.437	35
TJ-120-1.5-φ1.5	1.5 × 1.4	120	1.5	2.0	1.33	2.57	0 ≤ γ ≤ 1	−946.04γ6 + 3180.2γ5 − 4025.5γ4 + 2465.2γ3 − 750.71γ2 + 104.87γ − 18.829	−1211.4γ6 + 3171.7γ5 − 3358.4γ4 + 1818.2γ3 − 513.54γ2 + 68.932γ − 40.437	35
BE120-1CA	1.3 × 1.5	120	1.5	2.0	1.33	3	0 ≤ γ ≤ 1	−3854.5γ6 + 11,587γ5 – 13,468γ4 + 7665.3γ3 − 2207.1γ2 + 296.96γ − 31.658	−4298.8γ6 + 12,213γ5 – 13,660γ4 + 7582.7γ3 − 2152.9γ2 + 287.86γ − 53.577	35
BE120-3CA	3.1 × 1.8	120	1.5	2.0	1.33	5.13	0 ≤ γ ≤ 1	−768.64γ6 + 1774.6γ5 − 1512.8γ4 + 594.2γ3 − 107.78γ2 + 7.8592γ − 23.62	−2178.8γ6 + 5494.3γ5 − 5524.7γ4 + 2801.7γ3 − 740.48γ2 + 93.984γ − 48.701	35

Note: when drilling with a twist bit, h/D₀ = 1.15~1.20.

.

For the measured strain ${\epsilon }_x$, ${\epsilon }_y$, the formula and the first stage A and B corresponding to the aperture in the table are used to calculate the initial value of residual stress. According to the numerical range of the initial value in the table, the corresponding A and B are recalculated. The second calculation result is the modified residual stress value. When any one of the initial residual stress ${\sigma }_x$, ${\sigma }_y$ exceeds the application range of the first stage A and B, it is necessary to use the corresponding A and B values for the second calculation. Suppose the two values exceed at the same time. The corresponding A and B values should be selected according to the largest absolute value for the second calculation. The result obtained is the revised actual stress value. Generally speaking, the A and B grading methods only need to be calculated twice. When the plastic strain is not included in the measured strain, the initial calculation value is the real stress value, and there is no need for a second calculation.

2.3. Experimental Verification of Residual Stress Field

The LXRD high residual stress tester is used to detect the residual stress. The LXRD high-speed residual stress tester uses X-ray diffraction to measure the residual stress. Compared with the blind hole method, the most remarkable feature of the X-ray diffraction method is that it can achieve nondestructive testing.

The main principle of the X-ray diffraction method is that when the elastic strain occurs in the crystal, the distance between the crystal planes will change. This change will cause the diffraction lines to shift so that the strain can be calculated from the magnitude of the shift under stress. For objects with residual macroscopic residual stress, the elastic strain is usually uniform in a small volume range (for example, centimeter-level). Moreover, because there is no triaxial stress on the object’s surface, it can be regarded as the plane stress state. Therefore, according to the geometric relationship, the formula for calculating the residual stress can be deduced on the theoretical basis of the Prague equation and Hooke’s law: σ_x = K × M. In the formula, K is a constant related to the material’s elastic constant and the X-ray incident angle θ₀. M is a function of the angle Ψ and position 2θ_Ψ of the diffraction peak.

Firstly, the corresponding target material and coefficient are selected according to the test material in the measurement process. Two detectors are used to project X-rays onto the sample surface at different angles, and the scanning countermeasures the diffraction peak position. The error of straight-line and ellipse fitting is carried out according to the curve, and the slope M value is obtained. The experimental measurement principle-stress equation is as follows:

$$\begin{array}{c}{\epsilon }_{\varphi \phi }^{\left\{hkl\right\}}=S_1^{\left\{hkl\right\}}\left[{\sigma }_{11}+{\sigma }_{22}+{\sigma }_{33}\right]+\frac{1}{2}{S}_2^{\left\{hkl\right\}}{\sigma }_{33}{\cos }^2\psi +\frac{1}{2}{S}_2^{\left\{hkl\right\}}\left[{\sigma }_{11}{\cos }^2\varphi +{\sigma }_{22}{\sin }^2\varphi +{\tau }_{12}\sin 2\varphi \right]{\sin }^2\psi \hfill \\ +\frac{1}{2}{S}_2^{\left\{hkl\right\}}\left[{\tau }_{13}\cos \varphi +{\tau }_{23}\sin \varphi \right]\sin 2\psi \end{array}$$

(7)

In the experiment, to accurately verify the model as a whole, nine points such as those in Figure 11a were selected on two groups of docking test plates, respectively, and the calculation results were widely compared. The residual stress measurement process is shown in Figure 11b.

Then, we used the blind hole method to measure the residual stress using the YC-III static program-controlled resistance strain gauge and ZDL-II residual stress drilling device to test the residual stress of the butt joint plate structure using the blind hole method as shown in Figure 12. The strain flowers were pasted at the position measured by the original X-ray diffraction method. Then, the blind hole with a hole diameter of 1.5 mm and a depth of 2.0 mm was drilled using a drilling device, and the corresponding strain value was measured. The residual stress value was calculated according to the stress release principle. The calculation formula is as follows:

$${\sigma }_{1,2}=\frac{E\left({\epsilon }_1+{\epsilon }_2\right)}{4A}\pm \frac{E}{4B}\sqrt{{\left({\epsilon }_1-{\epsilon }_2\right)}^2+{\left[2{\epsilon }_2-\left({\epsilon }_1+{\epsilon }_2\right)\right]}^2}$$

(8)

$$\theta =\frac{1}{2}{\mathrm{tg}}^{-1}\frac{2{\epsilon }_2-({\epsilon }_1+{\epsilon }_3)}{{\epsilon }_3-{\epsilon }_1}$$

(9)

Then, according to the modified formula of the corresponding variable release systems A and B, the modified formula was obtained:

$$\mathrm{A} (\mu \epsilon /MPa)=-946.04{\mathsf{\gamma }}^6+3180.2{\mathsf{\gamma }}^5-4025.5{\mathsf{\gamma }}^4+2465.2{\mathsf{\gamma }}^3-750.71{\mathsf{\gamma }}^2+104.87\mathsf{\gamma }-18.829$$

(10)

$$\mathrm{B} (\mu \epsilon /MPa)=-1211.4{\mathsf{\gamma }}^6+3171.7{\mathsf{\gamma }}^5-3358.4{\mathsf{\gamma }}^4+1818.2{\mathsf{\gamma }}^3-513.54{\mathsf{\gamma }}^2+68.932\mathsf{\gamma }-40.437$$

(11)

The stress ${\sigma }_x$, ${\sigma }_y$of the measuring point in the direction of x and y are as follows:

$${\sigma }_{x,y}=\frac{{\sigma }_1+{\sigma }_2}{2}\pm \frac{{\sigma }_1-{\sigma }_2}{2}\cos 2\theta$$

(12)

The results of the two test methods are shown in Figure 13. It can be seen from the figure that the experimental results measured by the X-ray diffraction method and blind hole method are consistent [32,33].

3. Conclusions

In view of the importance of residual stress distribution and law to straight pipes in deep-sea mining, the problem of inaccurate measurement accuracy of the blind hole method is studied for the high residual stress caused by the welding of pipeline steel. In this study, a three-dimensional finite element model was established using finite element software to simulate the stress release environment of a blind hole in the stress release state of steel with different yield strengths in the calibration test of the blind hole method. The strain release coefficients under different applied stress ratios to specimen yield strength γ were studied. The variation law of elastic strain and plastic strain under different γ values analyzes the variation trend of the strain release coefficient with γ value. The distribution reason of the variation trend of the strain release coefficient with γ value of three different strain gauges was discussed, and the variation law of the strain release coefficient with γ value was obtained by fitting.

(1)

According to the comparison between the finite element calculation and the classical solution, it can be found that the strain release coefficient of steel with different yield strengths is consistent when the ratio of applied stress to specimen yield strength is the same.

(2)

The inflection point of the strain release coefficient curve is mainly when the applied stress reaches 0.4 ${\sigma }_s$, 0.7 ${\sigma }_s$, and 0.9 ${\sigma }_s$. When the γ value reaches 0.4, the strain release coefficients A and B begin to change. Before reaching 0.7, the plastic strain in both directions is zero. When the γ value reaches 0.7, the plastic compressive strain increases perpendicular to the applied stress. When the γ value is more significant than 0.75, the plastic strain increases rapidly, changing the strain release coefficient A and B slope. When the γ value reaches 0.9, the plastic tensile strain begins to occur in the applied stress direction. The strain values in both directions increase rapidly, resulting in a change in the curve slope of strain release coefficients A and B.

(3)

The data was extracted according to the position of different strain grids (the distance from the center of the circular hole was different). Finally, the correction of high residual stress was established.