J. Cent. South Univ. (2020) 27: 3163-3174

DOI: https://doi.org/10.1007/s11771-020-4538-0

Measuring internal residual stress in Al-Cu alloy forgings by crack compliance method with optimized parameters

DONG Fei(董非)^{1, 2}, YI You-ping(易幼平)^{1, 2, 3}, HUANG Shi-quan(黄始全)^{1, 2, 3}

1. Research Institute of Light Alloy, Central South University, Changsha 410083, China;

2. State Key Laboratory of High Performance Complex Manufacturing, Changsha 410083, China;

3. School of Mechanical and Electrical Engineering, Central South University, Changsha 410083, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: Measuring the internal stress of Al alloy forgings accurately is critical for controlling the deformation during the subsequent machine process. In this work, the crack compliance method was used to calculate the internal residual stress of Al-Cu high strength alloys, and the effect of various model parameters of crack compliance method on the calculated precision was studied by combining the numerical simulation and experimental method. The results show that the precision first increased and then decreased with increasing the crack range. The decreased precision when using a high crack range was due to the strain fluctuation during the machining process, and the optimized crack range was 71% of the thickness of forgings. Low orders of Legendre polynomial can result in residual stress curve more smooth, while high orders led to the occurrence of distortion. The Tikhonov regularization method effectively suppressed the distortion of residual stress caused by the fluctuation of strain data, which significantly improved the precision. In addition, The crack compliance method with optimized parameters was used to measure the residual stress of Al-Cu alloy with different quenching methods. The calculated results demonstrated that the distribution of residual stress was obtained accurately.

Key words: residual stress; crack compliance method; crack range; interpolation orders; Tikhonov regularization method

Cite this article as: DONG Fei, YI You-ping, HUANG Shi-quan. Measuring internal residual stress in Al-Cu alloy forgings by crack compliance method with optimized parameters [J]. Journal of Central South University, 2020, 27(11): 3163-3174. DOI: https://doi.org/10.1007/s11771-020-4538-0.

1 Introduction

Heat-treatable Al alloys, such as 2000 series alloys, 6000 series alloys, and 7000 series alloys, are widely used to fabricate forged components used in aerospace and aircraft industry for weight reduction fields [1-3]. These alloys belong to heat treatable alloy. Solution, quenching and artificial aging are usually applied to achieve higher strengths. However, residual stresses are inevitably introduced during the quenching process owing to different cooling rates in the sample, especially in large scale components. Thus, their dimensional stability and service performance were significantly deteriorated. A great deal of research effort is focused on how to reduce residual stress. Of course, it is important to measure the residual stress of the samples.

X-ray diffraction and drilling methods are usually used to measure residual stress [4-6]. However, they can only measure the residual stress at a lower depth from the surface. Compared with X-ray, the penetration ability of neutrons diffraction [7] is stronger and it is conducive to measure the internal residual stress. Unfortunately, the measurement time of neutron diffractometer is long and the neutron source construction cost is expensive. The crack compliance method [8, 9] has the advantages of simple measurement and is widely used in the measurement of residual stress inside the components.

The principle of the crack compliance method (CCM) is to introduce an increasingly enlarging slit from the surface of the component to release the residual stress and record the strains at the back surface. The stress was inversely calculated by combining the recorded strains and compliance function. VAIDYANATHAN et al [10] initially used the photoelastic coating method to measure the stress intensity factor at different depths of the crack in 1971, which are then used to calculate residual stress. However, it is difficult to put it in practice on account of its complicated calculation process. By 1998, instead of measuring the stress intensity factor with photoelastic technology, CHENG et al [11] used strain gauges to measure strains or displacements to simplify the test and calculation process. They firstly used the crack compliance method for the determination of residual stresses in aluminum alloy plates. Since then, many researchers focused on how to improve the calculation precision. TANG et al [12] studied the effect of interpolation functions (e.g., Legendre polynomials, power series, and Fourier series) on the calculation precision, and concluded that Legendre polynomials can minimize the calculation error and the mean square root of the uncertainty was about 2.797 MPa. LI et al [13] advised that Legendre polynomials above the 12 orders will produce excess oscillation, resulting in stress calculation distortion. However, the factors that affected the measurement accuracy of residual stress are very complicated, and systematic research is still lacked.

In this work, the effect of different parameters such as crack range, interpolation orders, and penalty factors on the measurement accuracy of residual stress was studied by combining numerical simulation and experiments. The flowchart of this research is shown in Figure 1. The solution-quenching process for the 2A14 Al alloy sample was simulated by a numerical simulation software Abaqus and the quenched residual stresses of the sample were obtained. Similar to measuring the strains in CCM, the strains of the sample with different cutting depths were then obtained by simulating the cutting process. Next, the residual stress was calculated by CCM using the strain with various parameters, and the parameters were optimized by comparing the calculated stress with simulated stress. Based on the optimized results, the CCM was used to calculate the internal residual stress of 2A14 Al-Cu alloy with different quenching process. In addition, the residual stress at the surface of the specimen was tested by X-ray diffraction method.

Figure 1 Flowchart of this research

2 Methods

2.1 Principle of CCM

The process of measuring residual stress using CCM is shown in Figure 2. To simplify the problem, this paper only studied the residual stress in the x-direction. The residual stresses at different crack depths are calculated by measuring the strain values at the specified point corresponding to the different crack depths. As shown in Eq. (1), it is assumed that the residual stresses σ_x,z(y) are evenly distributed in the x-z plane of the sample, and the residual stress only changes in the y-direction.

                       (1)

where A_i is the interpolation coefficient; P_i(y) is the interpolation function, which can be Power series,Fourier series, Legendre polynomial, or other polynomials [11-13]; n is the order of the interpolation function.

Figure 2 Schematic diagram of crack compliance method

In this paper, the Legendre polynomial was used as the interpolation function. Since the interpolation function is given, the process of calculating the σ_x,z(y) can be transformed into the process of calculating A_i. According to the superposition principle, the strains can also be expressed as a series expansion, as presented in Eq. (2):

                      (2)

where C_i(y) is crack compliance function, which is the strain recorded by strain gage when the crack grows to a length of y if stress with P_i(y) was applied to the sample, and the C_i(y) is written as C_ij in this work. The calculation of C_ij will be described in Section 2.2.

To reduce the calculation error, the least- squares method was used to fit the strain calculated from the compliance function and the measured strain data, as shown in Eq. (3):

         (3)

where ε_measured(y) is the measured strain at which the crack depth is y; and m is the number of strain data in the thickness range of the specimen. In the actual measurement process, m>>n. Hence, we can obtain the coefficient A_i in Eq. (1) by solving Eq. (4).

                       (4)

2.2 Calculation of crack compliance function

There are multiple ways to calculate the crack compliance function (C_ij), such as the physical method, fracture mechanics analysis, finite element method (FEM), and other numerical methods [14, 15]. In this paper, FEM was used to analyze and calculate the crack compliance function because FEM has the advantages of fast calculation and high accuracy.

Based on the plane stress assumption, a two- dimensional (x-, y-direction) finite element model of the test specimen was established. As shown in Figure 3, half of the sample was meshed. The crack was taken as the symmetry plane owing to the symmetry of the specimen. The width of the left grid of the model was 0.5 mm, which was exactly the width of the actual crack. The remaining part of the grid size was 1 mm×1 mm. An 8-node plane strain unit CPE8 was used. The freedom in the y-direction of the left sample was constrained, and the displacements at point A in both the x- and y- directions were also constrained. The elastic modulus of the Al alloy material was 81.5 GPa and the Poisson ratio was 0.406.

Figure 3 Diagram of mesh generation and application of boundary conditions

The Legendre polynomial P_i(y) with different orders was applied as initial residual stress along the symmetry line in the interior of the sample. To simulate the growth of crack, the “model change” was used to gradually remove the elements from the left side of the sample in the y-direction. Simultaneously, the displacement value of points A and B was recorded. The strain value can be calculated by Eq. (5):

                      (5)

where U_B and U_A were the displacement value at points B and A, respectively; L_gauge was the length of strain gauge. Consequently, we can obtain the strain value at point B as the crack growths with different orders of the Legendre polynomial.Figure 4 exhibits the crack compliance function for different Legendre polynomials.

Figure 4 Crack compliance function of Legendre polynomials:

2.3 Numerical simulation of quenching residual stress

2.3.1 FEM model

The numerical simulation software Abaqus was used to further study the difference between the calculated residual stresses and simulated residual stress. The dimensions of the original sample were 110 mm×100 mm×70 mm. As shown in Figure 5, only half of the sample was used to reduce the calculating time, and the dimensions of the model were 55 mm×100 mm×70 mm, as a consequence of the symmetry. The element type used in this model was an 8-node thermally coupled brick, trilinear displacement, and temperature element (C3D8T). The model was divided into two areas: the first section was located in the area of the left side of model with dimensions of grid being 1 mm×2 mm× 2.5 mm, and the second section was located in the remaining area with dimensions of 2.5 mm×2.5 mm×2.5 mm. The number of the elements was 33250, and the solution technique was the full Newton method. The thermal property parameters of the studied material, such as conductivities, specific heat capacity, elasticity modulus, and thermal expansion coefficient, are listed in Table 1. The density of the material is set constant at a value of 2800 kg/m³.

Figure 5 Numerical simulation model

Table 1 Thermal properties

2.3.2 Simulation process

Firstly, the solution-quench process of the sample was simulated by the Abaqus standard with coupled temperature-displacement analysis. The solution temperature of the model was 500°C, and the quenching temperature was 20 °C. During the quenching process, the two end surfaces (see Figure 5) were taken as the main heat-dispersing surfaces. Secondly, using the inheritance function to simulate the crack propagation process based on the model of the previous operation. The initial stress field of the model was the residual stress field calculated from the previous model. Consequently, the displacement value of point A can be extracted, allowing the strain values at point A at different depths to be determined. Thirdly, the CCM method with different parameters was then used to calculate the residual stress based on the above-measured strains.

2.4 Experimental

Two 2A14 Al alloys samples were taken from a hot-forged component with a chemical composition of 4.28 wt% Cu, 0.6 wt% Mg,0.81 wt% Mn, and 0.94 wt% Si. The sample dimensions were 110 mm×100 mm×70 mm, which was the same as the numerical simulation. The two end surfaces were taken as the main heat-dispersing surfaces, while the other four surfaces were encapsulated with about 20 mm-thick asbestos to avoid heat transfer. The samples were immediately quenched by using the immerse quench method (IQM) and spray quench method (SQM) with room- temperature water after solution treatment at 500 °C for 3 h, and the diagram of spray quench process is shown in Figure 6. The spray pressure was 0.3 MPa.

The wire electrical discharge machine (DK7735) was used to produce the cracks because the excess residual stress introduced by the method was very small. The strain gauge (BHF120-3AA) with dimensions of 5 mm×3 mm was used. Due to the large error of the residual stress at the surface of the sample calculated by the CCM, the residual stress at the surface of the sample was tested by X-ray diffraction. The detailed measure method was reported by ZHANG et al [16].

Figure 6 Diagram of spray quench process

3 Result and analysis

3.1 Simulated results

Figure 7(a) shows the residual stresses in the x-direction of the quenched sample with different depths. The results indicated that the residual stress changed from compressive stress to tension stress as depth increased. On account of the symmetry condition, the residual stress of the model was symmetrically distributed along the thickness direction. The stress at the surface point was -150 MPa and the stress at the middle point was 130 MPa.

As shown in Figure 5, the cutting process was simulated and the strains with different cutting depths were recorded for calculating the residual stress by CCM. Figure 7(b) shows the strains at point A during the crack propagation process. The results indicated that the strain value at point A increased with the extension of crack. The strain reached its maximum value when the depth of the crack was about 50 mm. Then the strain value at point A was gradually reduced, which was because the residual stress within the model was gradually released as the extension of crack. In the beginning, due to the high stiffness of the material, the release of residual stress only caused a lower strain at point A. And with the increase of residual stress inside the model and the decrease of model stiffness, the strain value of point A gradually increased. When the stress was completely released, the strain at point A finally decreased to a low value.

Figure 7 Simulated results:

3.2 Influence of crack range on precision of residual stress

The number of strain data directly affects the calculated results of residual stress because the computing process of CCM used the interpolation method [17-19]. It is obvious that the number of strain data was mainly determined by the length of cracks. Hence, we studied the influence of strain data on the precision of calculated stress by using different crack ranges (i.e., 50%, 65%, 71% and 93% of thickness). According to Ref. [12], when the depth of the crack was less than 5 times the diameter of the wire, the strain data were not valid. Consequently, we selected the strain data in Figure 7(b) with the length of crack larger than 5 mm, and Legendre polynomials with 2-10 orders were used.

Figure 8 shows the simulated stress and calculated stress by CCM with different crack ranges. The results indicated that the difference between the calculated residual stress and the simulated ones first decreased and then increased with the increase of crack range. When the crack range was 50% of the thickness of sample (i.e., 5-35 mm), the calculated results of the residual stress were extremely unstable, and only the results within the range of 32-35 mm were consistent with the simulation results, as shown in Figure 8(a). When the slit range increased to 65% (i.e., 5-45 mm), the consistent range between the CCM and simulation results was increased to 30-45 mm (see Figure 8(b)). To further expand the range of choice of strain data to 71% (i.e., 5-50 mm), the results were in well agreement with the simulation ones, with an exception for the results at the endpoints (see the blue circle in Figure 8(c)).

Figure 8 Residual stresses calculated by CCM using various crack ranges:

Figure 8(d) shows the calculated residual stress with a crack range of 93% (i.e., 5-65 mm). The results indicated that the calculated stress and simulated stress were very close within the range of 5-50 mm. However, the abnormal jitter occurred in the calculated residual stress curve at the depth of 57 mm, which resulted in a large deviation from the solution within the range of 50-65 mm (see blue circle). It can be explained as below, as shown in Figure 7(b), the strain gradually decreased and then increased with increasing the depth of crack within the range of 50-65 mm. As shown in Eq. (4), the magnitude of the strain and the slope of strain values directly affected the results of the interpolation coefficient, and thus the result of the residual stress was further affected, causing a large deviation at this point. It should be noted that the residual stress calculated using the CCM greatly deviated from the results obtained by the simulation method at the left and right endpoints. This was a consequence of CCM using the interpolation method. There was only one direction of the numerical points to limit the endpoints, increasing the probability of the deviations in the residual stress at the endpoints.

3.3 Influence of interpolation orders on precision of CCM method

At the same time, the calculation results were affected by the choice of interpolation orders of Legendre polynomial [20]. Hence, we studied the influence of the interpolation orders (e.g., 2-8, 2-9, 2-10, 2-11 and 2-12 orders) on the precision of the CCM method in this section, and the crack range was selected as 71% of thickness in all calculation process according to the optimized results in Section 3.2. Stress uncertainty was used to quantitatively evaluate the calculation derivation of residual stress with various orders. The calculation process is summarized as the following equations:

          (6)

where s_model,j(n) was the stress uncertainty at the crack depth of j when the interpolation order was n; was the average stress calculated using different interpolation orders; a=n-1, and b=n+1.

Figure 9 presents the calculation results with different orders. As shown in Figure 9(a), when the interpolation orders were 2-8, the resulting curve was very smooth, and there was a large deviation at the position where the residual stress curve appears to bend. Increasing the orders to 2-9 or 2-10, the calculated results coincided well with the simulated ones, as shown in Figures 9(b) and (c). However, when 2-11 orders and 2-12 orders of Legendre interpolation polynomials were chosen, the calculated residual stress versus depth curve appeared redundantly twisted. The stress uncertainty of 2-8, 2-9, 2-10 and 2-11 orders were 9.01, 8.39, 12.16 and 15.98 MPa, respectively. The results and analysis above indicated that choosing the appropriate interpolation order has a great influence on the accuracy of the calculation results. For this work, the best orders were 2-9 orders.

3.4 Influence of penalty factors on precision of CCM method

Owing to the compliance function matrix C was ill-conditioned, i.e., when small fluctuation of strain data occurs, residual stress calculated by CCM will change greatly. This can lead to larger calculation error (shown in Figure 9(d)) [21]. However, the slight change of strain was inevitable during the actual measurement. In this work, Tikhonov regularization method, based on the gradient of the stress curve and the curvature function, was used to solve this problem [22, 23]. This method can weaken the strain stochastic error and improve the stability of the calculation. As shown in Eq. (7), the calculation of interpolation coefficients was changed from Eqs.(3) to (7) due penalty function was added [24].

               (7)

where D_curve was the penalty matrix, which can be obtained by Eq. (8) and α was the penalty factor. The penalty factor was usually chosen from 0-1.

  (8)

Accordingly, we get the curvature matrix as below:

Figure 9 Residual stress obtained by different interpolation orders:

Figure 10 shows the calculated results obtained by selecting different penalty coefficients (i.e., 0.01, 0.3, 0.5 and 1). The results indicated that the Tikhonov regularization can effectively solve the calculation error of residual stress caused by the strain fluctuation, and α has a great impact on the calculated results. When the selected parameter α was 0.01, the role of punishment was not obvious, and there was still a large number of deviated data within the range of 50-65 mm (shown in Figure 10(a)). When the selected parameter α was too large (α=1), it will lead to the calculated results of the residual stress very smooth (shown in Figure 10(d)), and it is unable to reflect the residual stress curve characteristics. To achieve the best results, we concluded the optimized α as 0.3 or 0.5. Besides, it is noted that the calculated stability at the endpoints by the Tikhonov regularization was very good as compared to the conventional method.

Figure 10 Residual stress calculated from different penalty coefficients:

4 Application of CCM

The CCM method with optimized parameters was applied to calculate the residual stress of 2A14 Al alloy forgings with different quenching methods (i.e., IQM and SQM). Figure 11 shows the strain data with crack depth recorded by strain gauges. The strain trend was similar to the simulation result (Figure 7(b)) except in the starting region and ending region of cracks. Strain fluctuation was found owing to the vibration of molybdenum wire during the starting machine process. In addition, it was found that the strain values sharply increased when the cracks grew to the end surface, which was caused by the gravity of samples. Hence, the effective strain range was determined as 5-50 mm. Then, the internal residual stress of SQM was calculated by CCM with different parameters. It is found that the effect of crack range on the precision of stress was similar with the simulation result, i.e., when the slit range was 50% or 57% of the thickness of sample, serious distortion occurred; while more accurate calculation results were obtained when the crack range was 71% of the thickness of sample (see Figure 12). Additionally, severe stress fluctuation of residual stress was investigated when using high orders of interpolation polynomials (see Figure 13).

Figure 11 Strain versus depth curves of IQM and SQM measured by CCM

Figure 12 Residual stresses obtained using different crack ranges

Figure 13 Residual stress calculated by CCM with different interpolation orders

According to the above results and analysis, we concluded the optimized parameters for Al alloy forgings, i.e., the slit range was about 71% of sample thickness, and the interpolation orders were 2-9 orders. It should be noted that the Tikhonov regularization can greatly smooth the curves of stress caused by the recorded strain fluctuation. The internal residual stress curves are shown in Figure 14, and the penalty coefficient of IQM and SQM were 0.3 and 0.5, respectively. As shown in Figure 14, it was found that the residual stress symmetrically distributed in the sample, and the stress gradually changed from compressive stress to tensile stress. The maximum residual stress of IQM was larger than that of SQM due to a higher cooling rate during the quenching process. To obtain the complete residual stress curve inside the material, the residual stress on the surface of the material was obtained by using X-ray diffraction because the endpoint obtained by CCM was poor. The results show that the residual stress of IQM and SQM on the specimen surface was (-131.15±8.12) MPa and (-88.67±6.73) MPa, respectively. From Figure 14, it was found that the surface residual stresses tested by X-ray for the two samples were coincidence with the predicted results of CCM, indicating the reliability of the CCM.

Figure 14 Residual stress distribution in samples with different quenching method

5 Conclusions

In the present study, the influence of parameters of CCM, such as crack range, interpolation orders, and penalty coefficients, on the calculated precision of internal residual stress of Al alloy forgings was studied. And these parameters were optimized by using simulation and experimental methods. The following conclusions are drawn.

The precision of CCM was gradually increased when the crack range increased from 50% to 71% of thickness because more strain data were used. But serious distortion of residual stress occurred when the crack range was further improved to 91%, because of the dramatic changes in strain data. The optimized crack range was determined as 71% of the thickness of samples. The curve of residual stress was very smooth when using low interpolation orders of Legendre polynomial, while high orders can result in unnecessary distortion. Thus the appropriate orders were 2-9 orders. Tikhonov regularization method with penalty coefficients of α=0.3 or 0.5 can effectively regulate the distortion of residual stress caused by strain fluctuation. In addition, the calculated precision at the endpoints was improved.

The CCM with optimized parameters was applied to measure the internal stress of Al alloy forging with immersed quenching and spray quenching method. Typical residual stress distribution with external compressive stress and internal tensile stress was obtained. The surface stress test by X-ray was well consistent with the CCM predicted stress, which demonstrated that the CCM can successfully measure the internal stress of Al alloy forgings.

Contributors

The overarching research goals were developed by DONG Fei, YI You-ping, and HUANG Shi-quan. YI You-ping contributed to the conception of the study. DONG Fei and HUANG Shi-quan designed and performed the experiments. DONG Fei performed the data analysis and wrote the manuscript. All authors replied to reviewers’ comments and revised the final version.

Conflict of interest

DONG Fei, YI You-ping, and HUANG Shi- quan declare that they have no conflict of interest.

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(Edited by ZHENG Yu-tong)

中文导读

采用优化参数的裂纹柔度法测量铝铜合金锻件内部残余应力

摘要：准确测量铝合金锻件内部残余应力对于控制合金后续机加工变形具有重要作用。本文采用裂纹柔度法计算了铝铜合金锻件内部残余应力，结合实验和仿真的方法研究了裂纹柔度法模型参数对残余应力计算精度的影响。结果表明，随着引入裂纹深度的增加，计算精度先逐渐增加再逐渐降低。机加工过程中的应变数据波动导致残余应力计算精度降低，最优的裂纹深度范围为锻件厚度的71%。采用低阶插值阶数会导致残余应力曲线过于平滑，而高阶插值阶数会使残余应力曲线出现扭曲。吉洪诺夫正则法可有效抑制应变数据波动导致残余应力曲线计算失真，从而大幅提高计算精度。采用优化的裂纹柔度法测量了不同淬火方式铝铜合金锻件内部残余应力，计算结果表明使用该方法可以准确地测量合金内部残余应力。

关键词：残余应力；裂纹柔度法；裂纹范围；插值阶数；吉洪诺夫正则化

Foundation item: Project(51875583) supported by the National Natural Science Foundation of China; Project(zzyjkt2018-03) supported by the State Key Laboratory of High Performance Complex Manufacturing, China

Received date: 2020-04-13; Accepted date: 2020-08-25

Corresponding author: YI You-ping, PhD, Professor; Tel: +86-18874869619; E-mail: yyp@csu.edu.cn; ORCID: https://orcid.org/0000- 0002-7104-8293; HUANG Shi-quan, PhD, Associate Professor; Tel: +86-13548981584; E-mail: huangsqcsu@ sina.com; ORCID: https://orcid.org/0000-0001-7747-1802