1. Introduction

2. Simulation modeling description

2.1. Aging simulation modeling

2.1.1. Quenching Experiment and Residual Stress Testing Experiment of Aluminum Alloy Ring Parts

2.1.1.1. Thermal parameters of 2219 aluminum alloy material

The research object of this article is 2219 aluminum alloy, and the thermal physical properties of aluminum alloy materials are greatly affected by temperature. The numerical simulation process of TSR and TVSR also needs to consider the material's thermal physical properties and mechanical performance parameters that vary with temperature. According to literature [28,29], the average density of 2219 aluminum alloy is 2840 kg/m3, and the Poisson's ratio is 0.3. According to reference [30], the quenching temperature of 2219 aluminum alloy is 535 ℃. Therefore, the temperature range for the thermal properties of the material is between 20 ℃ and 700 ℃. The fitting curve of material properties related to temperature of 2219 aluminum alloy is shown in Figure A1 [27]. Figure A1 includes such thermophysical property parameters as elastic modulus, specific heat capacity, density, thermal conductivity, coefficient of thermal expansion and yield stress of 2219 aluminum alloy material.

2.1.1.2 Experimental Process Route for Quenching Aluminum Alloy Rings

The connecting part blank of the rocket fuel storage tank studied in this article is a circular part, with an outer diameter of 450 mm, an inner diameter of 366 mm, and a height of 105 mm. The 2219 sample aluminum alloy ring is placed in a heating furnace and heated from room temperature of 25 ℃ to 535 ℃ for insulation. The heating time is 120 minutes, the insulation time is 150 minutes, and the temperature is controlled within ± 3 ℃. After the insulation is completed, the sample is placed in a 25 ℃ water tank for cooling treatment, with a transfer time of less than 10 seconds, resulting in a large amount of residual stress generated by the temperature gradient caused by rapid water quenching. The quenching process route of 2219 aluminum alloy ring components is shown in Figure 1a.

a) Process route of 2219 aluminum alloy quenching ring, b) Quenching experimental equipment

In order to evaluate the numerical simulation results of stress aging, the initial residual quenching experimental stress is introduced into the stress aging numerical simulation process. Design an aluminum alloy ring quenching experiment based on Figure 1a aluminum alloy quenching process roadmap, using an electronic temperature control unit with timing function, which can control the temperature within ±3 ℃, accurately adjust the quenching temperature, and ensure the minimum temperature error between the experimental temperature value of the ring and the theoretically designed quenching process route. The on-site quenching experimental equipment and cooling pool are shown in Figure 1b.

2.1.1.3 Residual stress testing of aluminum alloy ring after quenching

In order to investigate the distribution status of residual stress in the ring blank and evaluate the magnitude of residual stress inside the ring, it is necessary to conduct residual stress testing on the experimental ring before numerical simulation of stress aging.

The initial residual stress is one of the main factors causing the deformation of the workpiece during processing. Due to the presence of residual stress in any part of the ring, precise multi point measurement is not yet feasible, and as of now, there is no universal and accurate residual stress testing technology due to the imperfect theory of residual stress testing for parts and the limitations of measuring equipment. The sample size of this experiment is relatively large. In order to more comprehensively and accurately reflect the stress distribution law of the ring rolled piece, 1 point was taken every 90 ° along the circumferential direction in the axial direction, and a total of 8 measurement points were taken. The actual photos of residual stress measurement points for 2219 aluminum alloy ring pieces at the experimental site are shown in Figure A2a.

Due to the fact that the measured component is a ring component, in order to evaluate the residual stress of the ring component as much as possible, combined with existing experimental conditions, the multi-point drilling method was used for residual stress measurement.

The drilling method is an early developed, low-cost, widely used, and most mature residual stress testing method. It has the advantages of high measurement accuracy, convenient operation, and cheap equipment, and has been determined as the standard residual stress measurement method by ASTM. The basic principle of the drilling method is to drill a small hole on the surface of the workpiece that generates residual stress. The stress in the vicinity of the hole is released, resulting in the corresponding strain. By measuring the magnitude of the strain, the average residual stress in the depth direction of the drilling can be obtained through calculation. The measurement principle is shown in Figure A2b.

The drilling method used in this article to measure the residual stress of 2219 aluminum alloy rings is based on the American Material Testing Standard E-837-13a [31]. As mentioned earlier, residual stress cannot be directly measured. Typically, the elastic strain generated by the initial release stress at the hole location is directly measured, and then residual stress can be calculated using mathematical relationships. In the experiment, firstly, according to the requirements of sticking strain gauge to measure residual stress by drilling method, stick strain rosettes, connect the measuring bridge circuit (this experiment uses a half bridge measuring circuit), and record the output value of the bridge when the hole is not drilled. The drilling device is used to drill holes at the drilling position to release the residual stress inside the workpiece. At this time, the resistance of the working strain gauge will change, and the output of the bridge circuit will change. Record the output result at this time to obtain the change value of the output voltage. The residual stress measurement system of the ring using the drilling method is shown in Figure 2a, and the on-site clamping is shown in Figure 2b.

When measuring, the measuring points should be polished and polished first to ensure a bright, flat, and scratch free surface, avoiding slight vibration of the pasted three-dimensional strain pattern during drilling and increasing measurement error. After the three-dimensional strain pattern is pasted, it is welded to the strain measuring instrument for wiring, and then the 2219 aluminum alloy experimental sample and drilling fixing seat are fixed to prevent the sample from slipping and swinging, and to avoid the phenomenon of drilling and milling cutter breakage. Through the built-in eyepiece, the fine-tuning drilling and milling cutter is located at the center of the three-dimensional strain pattern. The drilling method equipment uses high-pressure nitrogen gas cylinders to provide high-speed drilling and milling power for the drilling and milling cutter. Each measuring point measures a depth of 1.0 mm, starting from the surface and measuring residual stress every 0.1 mm. This means that each measuring point takes 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 mm in the radial depth direction of the ring as a group, totaling 8 sets of residual stress measurement results. According to the principle of drilling method, the measurement results are two main stress directions and main stress direction angles. Usually, the residual stress on the surface of the structure is in a plane stress state, and the two main stresses and main stress direction angles are unknown [32]. Therefore, it is necessary to use a three-dimensional strain pattern for measurement. As the residual stress test sample in this article is an aluminum alloy ring, uneven and smooth polishing will prevent the correct adhesion of the three-dimensional strain pattern, leading to significant measurement errors. Derive the strain values before and after drilling through mathematical formulas and measure the three-dimensional strain pattern ε₁, ε₂ and ε₃. Using strain to deduce stress, the calculation relationship is shown in equation (2-1).

(2-1)

where ${\sigma }_1$, ${\sigma }_2$——Two directions of principal stress；

$\theta$——The angle between the main stress ${\sigma }_1$ and the sensitive grid $R_1$；

$E$——Elastic modulus；

${\epsilon }_1$, ${\epsilon }_2$, ${\epsilon }_3$——Three direction strain value of three direction strain flower；

$A$, $B$——Strain release (correction) coefficient.

As the result value of residual stress of the ring measured by the drilling method is the reference result value in the plane Cartesian coordinate system. Therefore, it is necessary to transform the Cartesian coordinate system stress component into the polar coordinate stress component characterizing the cylindrical coordinate system of the ring piece [33] through the balance method of micro elements in elasticity. The conversion principle is shown in Figure 3.

According to the calculation program H-DRILL manual for residual stress in drilling, the recorded strain values will be ε₁, ε₂ and ε₃ After inputs, specify the type of three-dimensional strain pattern to calculate the principal stress in both directions σ₁, σ₂ and plane shear stress τ₁₂. Based on the stress situation of the triangular micro element and the geometric relationship of each side in Figure 3, the residual stress measurement results are transformed to obtain the radial, circumferential, and shear residual stresses of the ring at the measurement point. For the convenience of rapid calculation, it is rewritten in matrix form, and the results are shown in Equation (2-2).

(2-2)

where ${\sigma }_r$——Radial stress;

$\theta$——The angle between the main stress ${\sigma }_1$ and the sensitive grid$R_1$;

${\sigma }_{\theta }$——hoop stress;

${\tau }_{r\theta }$ and ${\tau }_{\theta r}$——shearing stress.

2.1.2. Stress relaxation test for aluminum alloy ring components

Before the stress relaxation test, solid solution treatment should be carried out in a resistance heating furnace, with a solution temperature of 535 ℃ and a insulation time of no less than 35 minutes. The potentiometer is used to control the furnace temperature, and the error is controlled within ±3 ℃; Room temperature water quenching, with a quenching transfer time of less than 35 seconds. Perform stress relaxation tests immediately after solution treatment.

Process standard stress relaxation test specimens according to the metal stress relaxation test standard, with a sample size of 4. Subsequently, stress relaxation tests were conducted on an electronic high-temperature creep endurance strength testing machine. After completing the sample preparation, stress relaxation tests were conducted on the RD-50 electronic creep endurance testing machine at the National Non ferrous Metals and Electronic Materials Analysis and Testing Center in Beijing, China. The equipment prototype is shown in Figure A3. In addition, the aging heat treatment temperature range of aluminum alloy materials is generally 120~200 ℃, while the artificial aging heat treatment temperature of 2219 aluminum alloy materials is generally around 175 ℃. After correctly installing the card, set the test temperature to 175 ℃ and the test time to 18 hours. Based on the stress measurement results of the blank ring, set four different stress levels to conduct stress relaxation tests under stress conditions of 50 MPa, 150 MPa, 250 MPa, and 350 MPa, and obtain the test data. After data processing, obtain the stress relaxation curves at each stress level.

There are many constitutive equations to describe the stress relaxation behavior of metals. Different empirical formulas are used to fit data to obtain constitutive parameters, such as logarithmic, exponential and Maxwell equations. Accurate stress relaxation equations are helpful in calculating stress relaxation at any initial stress and at any time, with the most important being exponential equations. Compared to exponential equations, delay functions have the characteristics of high accuracy, high applicability, and flexibility [34]. The delay function equation is shown in equations (2-3).

(2-3)

where $\sigma$-instantaneous stress;

$t$-Stress relaxation time;

${\sigma }_{\infty }$-Residual stress after prolonged stress relaxation;

$a_1$, $b_1$, ... , $a_n$, $b_n$-Determine the shape of the curve, indicating the speed of stress relaxation.

Based on actual data, this article uses quadratic and cubic delay functions to fit the stress relaxation curves of 2219 aluminum alloy under various experimental stress levels. The fitting results are shown in Table 1.

The curve fitting adopts an improved optimization algorithm Levenberg Marquardt, with an allowable error of 1E-6, and retains three decimal places of the stress relaxation curve fitting parameters. The experimental values and fitting results are shown in Figure 4. The correlation coefficients for stress levels of 50 MPa, 150 MPa, 250 MPa, and 350 MPa were 0.9976, 0.9933, 0.9996, and 0.9988, respectively. The closer the correlation coefficient of the fitting curve equation is to 1, the better the curve fitting effect.

The raw data and fitting results of the stress relaxation test shown in Figure 4. Based on experimental data and corresponding fitting curve equations, a set of nonlinear equations containing unknown creep parameters of C1~C4 materials was established. The creep results of 7075-T6 aluminum alloy material were obtained by solving the equations, as shown in Table 2.

2.1.3. Stress Aging Simulation Modeling of Aluminum Alloy Ring Parts

In order to study the stress control effect of thermal vibration composite on the internal stress of large aluminum alloy ring blanks, 2219 material was selected for rolling aluminum alloy rings in this paper, and the geometric dimensions of the rings were consistent with the actual situation. Apply ANSYS for finite element modeling and solving to complete the research on residual stress regulation and deformation control of 2219 aluminum alloy ring components. Select an 8-node Solid185 element for grid division, which has the ability of plasticity, hyperelasticity, stress stiffening, creep, large deflection, and large strain. Generally, the vibration response of small parts adopts a platform type excitation mode, and the parts are fixedly connected to the excitation platform. In order to reduce the impact of external constraints on the calculation results of the numerical simulation model, the platform is considered as an elastic constraint. In order to avoid external constraints affecting the simulation calculation results, Song et al. [35] implemented the constraint points at the bottom of the workpiece through a large number of spring elements, which not only ensured the convergence of the calculation but also limited the rigid body displacement of the workpiece in three directions, which is closer to the actual surface contact. Therefore, the surface nodes of the ring are covered with spring element Combin14 with smaller element stiffness, with a stiffness value of 1E-4 mm/N.

Similarly, before numerical simulation using thermal or VSR methods, modal analysis is required to obtain the vibration frequencies and modes of each order of 2219 aluminum alloy ring, and the first 10 modes are solved. The results are shown in Figure 5. From Figure 5, it can be obtained that the first natural frequency (i.e. the seventh natural frequency in this article) of the ring is the excitation frequency of the 2219 aluminum alloy ring, with a frequency amplitude of 619.32Hz. The relative total deformation results of the seventh mode are shown in Figure A4a, and for the ring, the vibration has symmetry.

Most vibration relief equipment uses a rotating eccentric mass, which causes forced vibration of the components under the rotation of the eccentric wheel. The vibration equation of the components is shown in equations (2-4) [36].

(2-4)

where $X$- vibration displacement;

$M$- Total vibration mass;

$m$- Rotor eccentricity mass;

$e$- Eccentricity;

$\lambda$- The ratio of vibration frequency k to natural frequency k₀;

$Y$- Damping ratio.

From the above analysis, it can be seen that at the natural frequency of the component, in equations (2-4) λ= 1. From this, the vibration equation amplitude A shown in equation (2-4) can be obtained as shown in equation (2-5).

(2-5)

In general, loading at the peak of a component is more likely to reach the yield limit of the material, thereby reducing residual stress. This article selects the relative total deformation of the surface nodes on the ring based on the 7th order vibration mode of the ring, as shown in Figure A4b, and then multiplies it by a proportional coefficient p to replace the amplitude A. Considering the vibration situation of wave peaks and valleys, which is used to achieve the numerical simulation process of VSR, the vibration equation in this paper is shown in equations (2-6).

(2-6)

where $p$- scale factor;

$\left\{u_0\right\}$- The relative total deformation of the 7th order vibration mode node on the upper surface of the ring;

$f$- Hz 7th natural frequency, unit: Hz;

$t$- Time, in units of s.

Through the TVSR numerical simulation and experiment of 7075 aluminum alloy plate, the research results show that the finite element model considering stress relaxation effect and bilinear strengthening criterion has good consistency with the experimental results [13]. The TSR creep constitutive equation of 7075 aluminum alloy material and the bilinear strengthening criterion (BISO) of cyclic stress and strain obey [37, 38] are taken as the numerical simulation constitutive model, and the transient dynamic numerical simulation analysis is carried out in the form of equation (2-6) in the finite element software to realize the numerical simulation of thermal vibration combined aging process.

During the TSR process, due to stress relaxation and the reduction of elastic modulus and yield limit, the residual stress inside the part is released and homogenized. In finite element numerical simulation, the implicit creep equation with strain hardening can be used to simulate the stress relaxation process. Through the stress relaxation test of 7075-T6 aluminum alloy in Section 2.1.1, the test data can be fitted and a set of nonlinear equations can be solved to obtain the material creep constitutive parameters shown in Table 2.

Therefore, using the ANSYS APDL script language, the modeling process of the thermal mechanical coupling finite element model for the TVSR process of 2219 aluminum alloy ring samples is as follows:

(1) The unit type is defined as Solid185, and the material is 2219 aluminum alloy. The chemical composition of the material is shown in Table 3. Then, by consulting literature [27], some relevant mechanical parameters of 2219 aluminum alloy within the range of 25-200 ℃ were obtained. Adopting bilinear isotropic hardening model as the material plastic constitutive model.

(2) Establish outer diameter 450× internal diameter 366× high 105 mm circular ring geometry. The diameter and height of the inner and outer rings of the finite element model are consistent with the size of the sample in the experiment. The geometric modeling is divided into 40 layers along the height direction and 12 layers along the radial direction. The grid is generated by sweeping method, which is convenient to extract the residual stress results in different directions and analyze the aging effect.

(3) Establish a spring element with a lower normal stiffness around the solid element, with a normal stiffness set at 10mm/N to constrain displacement while allowing the specimen to deform freely.

(4) Based on the stress measurement test results, define the initial residual stress field in layers using the "inistate" command.

(5) Define transient thermal loads. Apply heating load according to the temperature time curve measured in the experiment. Defined periodic displacement boundary conditions and simulated the vibration of the workbench and specimen. The finite element model was solved through transient analysis and the data was processed during post-processing. The numerical simulation process of 2219 aluminum alloy ring TVSR is shown in Figure 5.

Therefore, according to the numerical simulation process of TVSR in Figure 5, the residual stress evolution during the stress release process can be obtained by solving the established finite element model of TVSR.

3. Results and Discussion

3.1. Correlation analysis between initial residual stress experiment and numerical simulation

The Pearson correlation coefficient is commonly used in statistics to describe whether there is a correlation between continuous variables and the magnitude of the correlation between variables. Although this article only used the drilling method to obtain the residual stress at 8 different drilling depths of the ring, the residual stress varies continuously on the surface and inside of the ring, and the measurement results can be considered both continuous and relevant. In finite element analysis, there are often errors between numerical simulation results and experiments, let alone measurement errors in the experiments themselves. Therefore, to characterize the correlation between the experimental measurement results of the initial residual stress of the ring and the numerical simulation results of the initial residual stress, the correlation coefficient (R) of the correlation concept in statistics can be used to measure [39, 40]. The correlation expression is shown in equations (3-1). The approximate calculation formula for Pearson correlation coefficient in this article can conveniently implement equation (3-1), and the calculation formula is shown in equation (3-2).

(3-1)

(3-2)

where $n$——Number of drilling depths;

$x_i$——Initial residual stress experimental data;

$\overline{x}$——Average of data of $x_i$;

$y_i$——Numerical simulation data of initial residual stress;

$\overline{y}$——Average of data of $y_i$.

When |R| approaches 0, it indicates that there is no correlation between experimental data and numerical simulation data, and the correlation becomes weaker; Generally, when | R | is above 0.5, it indicates a direct linear correlation between the data. The closer it is to 1, the stronger the data correlation. This article takes experimental results and numerical simulation data of drilling depths of 0.2, 0.4, 0.6, 0.8, and 1.0 mm at each measurement point, and calculates residual stress correlation coefficients in different directions based on the approximate calculation formula of Pearson correlation coefficients. The results are shown in Figure 9.

From Figure 9, it can be seen that the strong correlation in three directions of each measurement point accounts for over 37.5%, and the moderate correlation accounts for over 62.5%. Although a higher value of |R| does not necessarily indicate a higher accuracy of the numerical simulation model containing initial residual stress, the correlation between the measured residual stress in the overall experiment of 2219 aluminum alloy ring and the numerical simulation model shows a moderate to strong correlation between the initial residual stress in three directions. This indicates that the numerical simulation model of 2219 aluminum alloy ring containing initial residual stress can accurately reflect the size and distribution of residual stress inside the actual ring.

3.2. Numerical simulation results and analysis of residual stress field during stress aging process

This chapter takes the ring formed from high-strength 2219 aluminum alloy material as the research object, and establishes a numerical simulation model for the stress aging process. Based on temperature related material parameters and fitted creep constitutive parameters, the transient dynamic analysis of the complete method is used to solve the displacement, strain, stress, and force of the structure over time under the combined action of transient dynamic load and temperature load. Develop an APDL script to explore the effects of VSR, TSR, and TVSR aging on the homogenization and regulation of residual stress inside 2219 aluminum alloy rings. The numerical simulation process routes for VSR, TSR, and TVSR aging are shown in Figure 10.

In order to intuitively compare the numerical simulation results of TSR, VSR and TVSR, the X direction, Y direction, XY direction and the maximum equivalent residual stress of the ring under the cylindrical coordinate system are extracted respectively. The results are shown in Figure 11. To verify the simulation effect of TSR, VSR, and TVSR aging techniques on reducing initial residual stress in the blank. Calculate the residual stress relief rate in each direction before and after aging using equations (3-3) η, the results of the maximum residual stress relief rate in each direction are shown in Figure 12.

(3-3)

where ${\sigma }_{initial}$- Initial residual stress;

${\sigma }_{relief}$- Residual stress after aging.

From Figure 11, it can be seen that TSR, VSR, and TVSR can all reduce the three-dimensional residual stress of 2219 aluminum alloy rings. Among them, TSR and TVSR have a significant effect on regulating and reducing residual stress inside the ring, while VSR has a less effective regulation and homogenization effect than TSR and TVSR.

The three types of aging TSR, VSR, and TVSR have different effects on the homogenization and regulation of residual stress inside the ring. The maximum equivalent stress relief rates of VSR, TSR, and TVSR are 52.8%, 80.6%, and 82.2%, respectively. The relief effect TVSR>TSR>VSR indicates that VSR, TSR, and TVSR can effectively reduce and homogenize the residual stress inside the ring. Research has shown that the limited effectiveness of VSR is due to the dynamic stress generated by external excitation superimposed on the compressive stress on the material surface, making it difficult to reach the yield limit of the material. Based on Figure 11 and Figure 12, it can be seen that the fact that VSR affects the stress release of ring components indicates that due to the movement and change of internal grains under external loads, there is local micro yielding of the material on the surface. Therefore, the ability of VSR homogenization to regulate residual stress inside the ring is not as good as TSR and TVSR.

In addition, from Figure 12, it can be seen that the elimination rate of residual stress inside the control ring of TSR and TVSR is not significantly different in each direction, and the effect of reducing the ability of aging control is roughly equivalent. Therefore, in order to improve the ability of TVSR regulation and homogenization of residual stress inside the ring, the next chapter will conduct optimization simulation for identifying key parameters of thermal vibration, and obtain the optimal TVSR process parameters and aging plan.

4. Conclusion

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