Riveting Quality Improvement Mechanism of 2A10 Aluminum Alloy with Compound Feed Rates

Abstract

The riveting process is conventionally performed at a constant feed rate, overlooking the distinct deformation mechanisms inherent in its successive stages. This study introduces a novel compound feed rate approach to enhance the riveting quality of 2A10 aluminum alloy countersunk head rivets. A three-dimensional finite element model, validated experimentally, was developed to simulate the riveting process, segmented into three stages: free upsetting, hole wall interference, and driven head formation. An orthogonal experimental design was employed to investigate the effects of varying feed rates (1, 5, 10 mm/s) within these stages on key quality metrics: interference distribution, uniformity, and driven head geometry. Results demonstrate that increasing the feed rate reduces average interference but increases the driven head diameter, revealing a stage-dependent influence. A multi-objective optimization framework, integrating gray relational analysis with the entropy weighting method, was applied to balance these competing objectives. The optimal compound feed rate scheme of 10-1-10 mm/s (for the three stages, respectively) was identified. This optimized scheme improved interference uniformity by 1%, increased the critical shank-end interference (Point H) by 10.9%, and enhanced driven head dimensions compared to conventional constant-rate riveting.

1. Introduction

In modern aircraft manufacturing, aircraft assembly is of the utmost importance. An airplane typically comprises numerous components, and the assembly of these components inevitably involves a diverse range of connection techniques, including riveting, screwing, welding, gluing, and others. Bolted joints, for instance, are plagued by high manufacturing costs, while glued joints suffer from unstable quality. Due to these drawbacks, riveting has emerged as the most crucial type of connection, particularly when it comes to joining metal components like the skin, partition frame, and longitudinal beam [1,2,3].

Although riveting technology is widely applied in aircraft assembly, the selection of drilling and riveting process parameters has a significant influence on the deformation mechanism of riveting and the mechanical properties of the riveted joint. Good mechanical properties of the riveted joint are indispensable for guaranteeing aircraft safety in service [4]. As a vital segment of riveting technology, pressure riveting connects components via pressing fasteners such as rivets or rivet nuts into the workpiece [5,6]. In this study, the rivet and panel materials are 2A10 and 2A12 aluminum alloys, respectively. This combination is representative of aerospace applications. The 2A12 aluminum alloy is a hard, high-strength aluminum commonly used for critical load-bearing structures such as skins and frames due to its excellent strength-to-weight ratio. Meanwhile, 2A10 aluminum alloy, belonging to the Al-Cu series of hard aluminum alloys, is widely used as a rivet material in aerospace. It is typically subjected to aging treatment to achieve a favorable combination of strength and ductility, making it suitable for the plastic deformation required during the riveting process.

The strain rate is an essential factor affecting the plastic deformation of materials. In pressure riveting, the pressure feed rate, as a key process parameter, significantly impacts the strain rate. At low feed rates, the material surrounding the rivet hole deforms gradually. The plastic deformation is uniform, and the material flows in an orderly manner around the rivet shank. This facilitates better hole-filling and more consistent joint formation. The slow rate allows the material sufficient time to adapt to the rivet, ensuring a stable deformation process. As the feed rate increases, the deformation rate of the material accelerates. This gives rise to rapid yet less uniform plastic deformation. Insufficient time for the material to flow smoothly around the rivet causes local stress concentrations, potentially resulting in micro-cracks or defects near the rivet hole. Conversely, at very high feed rates, the inertia of the riveting tool and the workpiece becomes significant. These inertia forces introduce additional vibrations and impact effects during the process. Such dynamic effects severely disrupt the deformation mechanism, resulting in highly non-uniform deformation and potentially jeopardizing the integrity of the joint [7]. These process-induced variations in deformation directly manifest as critical differences in the final joint’s mechanical performance, residual stress state, and micro-structural characteristics.

The study by Wang et al. [8] pointed out that average interference increases with the riveting die feed displacement and decreases with the increase in the riveting die feed rate, a parameter that also critically influences the residual stress field and fatigue crack propagation resistance of the joint [9,10,11]. Yu’s study [12] on the influence of electromagnetic riveting loading rate on TA1 rivets deformation concluded that an excessive loading rate is not conducive to the formation of rivets and increases the chance of cracking, a phenomenon linked to adiabatic heating and alterations in material hardness; on the other hand, a lower loading rate is beneficial to the rivet deformation process by enhancing heat transfer to a larger area, delaying the deformation zone from cracking, and boosting grain refinement, which improves the deformation of rivets. Silvaye [13] investigated the deformation behavior of commercially available MS20426AD3-5 countersunk rivets for the aerospace industry. The flow curves of the AA-2117-T4 aluminum alloy did not show a significant strain rate dependence, but significant softening of the compressed samples due to adiabatic heating caused by deformation was observed at the highest test rate, underscoring the complex interplay between strain rate and thermal effects on material state. However, the samples deformed at low test rates showed no signs of macroscopic fracture. In contrast, local fractures occurred at medium and high test rates, highlighting the direct impact of deformation parameters on joint integrity under mechanical load [2,14].

The failure of riveting joints in tensile and shear tests has been extensively studied in recent years. These studies highlight that joints are often subject to complex stresses under operational loads, with failure typically occurring due to a combination of shear and tensile forces. A recent study [15] investigated the performance of riveted joints under tensile and shear stresses and found that joint strength is highly sensitive to the rivet material, geometry, and installation process. Additionally, Chen [15] demonstrated that rivet head design and the interference fit between the rivet and the workpiece significantly affect the joint’s performance in both shear and tensile testing.

Furthermore, the measurement of residual stresses in riveted joints has become an essential topic for evaluating the long-term performance of riveted connections. Xu [16] employed X-ray diffraction to measure the residual stress distribution in riveted aluminum joints and found that high residual stress concentrations around the rivet holes could compromise the joint’s fatigue resistance. Their study concluded that optimizing the riveting process could help reduce these residual stresses and improve the overall performance of the joint.

The changes in rivet material hardness during the riveting process also play a critical role in joint integrity. Studies such as those by Wang [17] have demonstrated that excessive deformation rates lead to adiabatic heating, which can significantly alter the hardness of rivet materials, particularly those made from aluminum alloys. This alteration in hardness has been linked to an increase in crack formation and a decrease in the overall strength of the joint. Conversely, a slower deformation process facilitates more uniform material flow, reducing the likelihood of brittleness and promoting grain refinement, which enhances the rivet’s ductility and the joint’s resistance to failure.

However, existing studies [18] on the precise influence of the feed rate on riveting quality remain insufficient in three key aspects. First, the riveting process is generally carried out at a constant feed rate, ignoring stage-dependent deformation mechanisms (e.g., the free upsetting stage, hole wall rivet bar interference stage, and rivet-driven head formation stage). Second, the trade-off among rivet-driven head size, interference value, and interference uniformity remains poorly quantified. Blindly pursuing a large interference value will reduce the rivet-driven head size and the occurrence of cracks around the hole. On the other hand, if the interference value at the end of the rivet shank is too small, it will affect interference uniformity. Third, although gray relational analysis (GRA) has been applied to process optimization, no study has integrated GRA with entropy weighting to optimize compound feed rates across different stages.

This study addresses these gaps by:

(1)

Decomposing the riveting process into three dynamic stages and establishing a stage-dependent Finite Element (FE) model with Johnson–Cook plasticity [19,20].

(2)

Quantifying the nonlinear correlations among driven head diameter, average interference value, and uniformity under compound feed rates.

(3)

Proposing a novel-entropy weighting hybrid optimization framework to derive the optimal compound rate (10-1-10 mm/s). This approach acquires the optimal parameter combinations for the feed rates at various stages, thereby enhancing the riveting quality.

This approach surpasses traditional single-stage optimization by integrating process decomposition, multi-objective optimization, and experimental validation, providing a new paradigm for high-precision riveting in aerospace manufacturing.

2. Experiments and Methods

2.1. Experiment Work

An automated drilling-riveting system was employed to process 120° countersunk head rivets. The system deformed 2A10 aluminum alloy rivets (tensile strength: 310 MPa) (Noah Aviation Standard Parts, Jinan, Shandong, China) within 2A12 aluminum alloy panels (tensile strength: 470 MPa) (Shanghai Hechuan Metal Materials, Shanghai, China), with both strengths certified per GB/T 228.1-2021 [21]. The joint geometry, illustrated in Figure 1a, consisted of a rivet with a 2.5 mm shank diameter and 7 mm length, and a panel hole with a 2.6 mm diameter and 0.65 mm countersink depth, resulting in a pre-riveting clearance fit. The material behavior was characterized using the Johnson–Cook model (parameters in

Table 1. (a) Chemical composition of 2A12 and 2A10 aluminium alloy materials (%) (Reprinted from Ref. [22]); (b) physical and mechanical properties of 2A12 and 2A10 aluminium alloy materials (%) (Reprinted from Ref. [23]).

(a)
Grade	Si	Fe	Cu	Mn	Mg	Ni	Zn	Ti	Fe + Ni	Al
2A12	0.50	0.50	3.8~4.9	0.30~0.9	1.2~1.8	0.1	0.30	0.15	0.50	Balance
2A10	0.25	0.20	3.9~4.5	0.30~0.50	0.15~0.30	-	0.1	0.15	-	Balance
(b)
Grade	Density (g/cm3)			Shear Modulus (GPa)		Elastic Modulus (GPa)			Poisson’s Ratio
2A12	2.8			26		68			0.33
2A10	2.8			27		69			0.31

) [22,23]. Based on this setup, a compound feed rate strategy (1, 5, and 10 mm/s for the free upsetting, hole wall interference, and driven head formation stages, respectively) was implemented to optimize riveting quality and efficiency.

The automated system provides high-precision motion control for accurate formation of both the countersunk and driven heads. As shown in Figure 1b, the riveting process consists of three sequential steps:

Step 1: Pre-feed of the upper rivet die. In this stage, the rivets were fed into the hole; the lower rivet die remained stationary, while the upper rivet die moved according to the set riveting displacement and rate to press the countersunk part of the rivet.

Step 2: In this stage, the lower riveting die starts loading while the upper riveting die maintains pressure. The riveting force–displacement history was recorded by the system’s load cell for subsequent comparison with simulation results. This stage concludes when the load reaches the preset value.

Step 3: Post-forming disengagement involves coordinated withdrawal of both upper and lower dies. This completes driven head formation and concludes the automated riveting cycle.

The specimens were all riveted using an automatic drilling and riveting system developed by Shenyang University of Technology, as illustrated in Figure 2. Both the rivet and panel materials consist of aluminum alloys—specifically 2A10 for the rivet and 2A12 for the panel, the latter having a thickness of 2 mm.

2.2. GRA Coupled with an Entropy Weighting Method

The riveting quality in this study is quantitatively defined by multiple objectives, which serve as the optimization goals. These objectives are as follows: The overall average relative interference (δ_a) of the rivet shank should be maximized within the recommended optimum range of 1.5% to 3.0% for aluminum alloy-riveted joints to ensure better mechanical properties. Thus, δ_a is considered a “larger-the-better” objective. Regarding interference uniformity, the distribution of interference along the rivet shank decreases from the driven head end (Point A) to the countersunk head end (Point H). The relative interference at Point H (δ_h) should be as close as possible to 3.0% to improve uniformity, while the relative interference at Point A should not be excessively large to avoid inducing cracks around the hole. Therefore, δ_h is a “larger-the-better” objective, and δ_a at Point A is treated as a “smaller-the-better” objective.

The coefficient of standard deviation (V) of the relative interference should be minimized to ensure a consistent and uniform load transfer path, making Va “smaller-the-better” objective. In terms of rivet head geometry, a larger average diameter (d_a) of the rivet-driven head is preferred to improve mechanical properties, and d_a is thus a “larger-the-better” objective. Furthermore, as the lower end of the rivet-driven head (d_b) directly contacts the plate, increasing its diameter within a reasonable range helps distribute residual stress more favorably. Therefore, d_b is also treated as a “larger-the-better” objective.

These interconnected and sometimes conflicting criteria necessitate the development of a systematic multi-objective optimization framework, which is elaborated in the following sections.

A pressure riveting simulation model was established to investigate the performance responses of different feed rates at various stages. A full-factorial experimental design was constructed, where each set of compound feed rates was assigned to the parametric simulation model. Pressure riveting simulation analyses were conducted separately for each combination, and the numerical results for various performance responses were extracted.

The gray relational analysis (GRA) method was then applied to normalize the results from the full-factorial experimental design, a step known as “grey relational generation.” GRA provides specific normalization methods for different types of target responses: “larger-the-better,” “nominal-the-best” (closer to a specific value T), and “smaller-the-better.” Calculations were performed on the “grey relational generation” results to obtain the gray relational coefficient (GRC) for each experimental group.

Based on the calculated GRC values, the entropy weighting method was used to assign weights to each performance response, thus obtaining the weight values for each response. Finally, the gray relational grade (G) was calculated by multiplying the GRC by its corresponding weights, as shown in Equation (1). The gray relational grades of each experimental group were calculated accordingly. The design scheme with the highest gray relational grade represents the optimal comprehensive performance:

$$\begin{gathered} G={\sum }_{k=1}^n{\beta }_k\gamma \\ {\sum }_{k=1}^n{\beta }_k=1 \end{gathered}$$

(1)

In Equation (1), γ represents the GRC; β_k represents the weight value of the k-th target response; and n represents the number of target responses. The compound feed rate that maximizes the value of the gray relational grade is selected as the optimal design parameter.

3. Simulation Study

3.1. Modeling

To illustrate the plastic deformation process, a three-dimensional finite element model was created, and the model could simulate the 120° countersunk head rivet’s riveting process. Consistent with the experiment, the thickness of the panel is 2 mm, and the materials of the rivet and panel are 2A10 aluminum alloy and 2A12 aluminum alloy. Considering the plastic deformation of the rivet and panel during the riveting process, the Johnson–Cook material constitutive model, as shown in Equation (2), was adopted.

$$\sigma =\left[A+B{\epsilon }^n\right]\left[1+C\mathrm{l}\mathrm{n}{\displaystyle \frac{\dot{\mathsf{\epsilon }}}{{\dot{\mathsf{\epsilon }}}_0}}\right]\left[1-{\left({\displaystyle \frac{T-T_r}{Tm-T_r}}\right)}^m\right]$$

(2)

where $\sigma$ is the Von Mises flow stress; $\epsilon$ is the plastic strain; $\dot{\epsilon }$ is the actual strain rate$;{\dot{\epsilon }}_0$ is the static tensile strain rate; $T$ is the temperature of the specimen; $Tr$ is the reference temperature, for which the default is room temperature; T_m is the material melting point of the specimen; A is the yield stress; and B, C, m, and n are all parameters related to the material properties of the specimen. The temperature term of Equation (2) can be neglected, using $20°\mathrm{C}$ as the reference temperature and $0.01{\mathrm{s}}^{-1}$ as the reference strain rate, and then Equation (1) simplifies to Equation (3):

$$\sigma =A+B{\epsilon }^n$$

(3)

In order to obtain the real stress–strain curves of two materials, the 2A12 aluminum plate and 2A10 rivet, in the study of mechanical properties, tensile tests on the 2A12 aluminum plate and compression experiments on the 2A10 rivet with the help of a tensile testing machine were carried out. The fitting curves of the Johnson–Cook model parameters for two aluminum alloy materials, 2A12 and 2A10, are shown in Figure 3.

The figure shows that the Young’s modulus of 2A12 and 2A10 aluminum alloys are 71.60 GPa and 68.99 GPa, respectively.

The initial yield stresses A were 320.78 MPa and 247.38 MPa, the strain-hardening moduli B were 694.256 MPa and 708.0345 MPa, and the strain-hardening indices were 0.6344 and 0.2658, respectively.

Fitting data can be used to obtain the C value at each strain rate by selecting a reference strain rate to be substituted for the yield strength at different strain rates. In this paper, the strain rate sensitivity factor of 2A12 hard aluminum alloy, which is also known as the C value, is 0.001, obtained using the stress–strain curve fitting at different strain rates at room temperature, obtained by Zhang Wei [23] of the Harbin Institute of Technology through the Hopkins compression bar test. The strain rate sensitivity factor of 2A10 aluminum alloy (C value) is 0.0083, obtained by Sanjeev NK [24] through finite element simulation.

The Johnson–Cook plasticity model for the hard aluminum alloy 2A12 is

$$\mathsf{\Sigma }=\left[320.78+694.256{\mathsf{\epsilon }}^{0.6344}\right]\left[1+0.001\mathrm{l}\mathrm{n}{\displaystyle \frac{\dot{\mathsf{\epsilon }}}{{\dot{\mathsf{\epsilon }}}_0}}\right]$$

(4)

The Johnson–Cook plasticity model for aluminum alloy 2A10 is

$$\sigma =\left[247.38+708.0345{\epsilon }^{0.2658}\right]\left[1+0.0083ln{\displaystyle \frac{\dot{\epsilon }}{{\dot{\epsilon }}_0}}\right]$$

(5)

In this study, finite element analysis was conducted using ABAQUS (Version 6.14-4), a commercial solver optimized for large-deformation metal forming. Three-dimensional modeling was essential to capture the asymmetric deformation of 120° countersunk head rivets, where the driven head formation and radial expansion of the rivet shank introduce complex 3D stress states. Unlike 2D axisymmetric models, the 3D approach with C3D8R elements accurately resolves non-uniform contact pressures and strain rate heterogeneity across stages (free upsetting to driven head formation). All parts in the model were defined as deformable bodies, except that river dies were defined as analytical rigid bodies. The grid of the riveting die was divided into 1632 elements and 2093 nodes. The mesh of the upper and lower plates was set to transition from dense to sparse from the hole to the edge. The grid of the top plate was divided into 12,544 elements and 14,760 nodes. The grid of the bottom plate was divided into 15,020 elements and 17,764 nodes. As for the rivet, 14,3832 elements and 15,2233 nodes were used for modeling.

To handle severe mesh distortion during large plastic deformation, an adaptive meshing (ALE) technique was employed. Automatic remeshing was triggered when the element geometric distortion metric exceeded a critical value of 0.8 (the default in ABAQUS). A swept-meshing technique was subsequently applied to the deformed regions, maintaining a minimum element size of 0.05 mm to ensure accuracy in capturing high strain gradients while preventing numerical divergence. The interactions within the model were simulated using the general contact algorithm in ABAQUS. This approach automatically manages all potential contact pairs, including those between the rivet and the panels, the rivet and the dies, and between the panels themselves. A global “Hard” contact property was defined for normal behavior, coupled with a constant Coulomb friction coefficient of 0.18 for all tangential interactions [25]. This simplified yet consistent contact definition is appropriate for the current study, as it ensures that the comparative analysis of feed rate effects is conducted under uniform boundary conditions. The key point of the FE model is how to set the riveting parameters to study the effect of parameters on formation quality. In this model, the motion of the rivet die was controlled by setting the displacement value and the load step time. This study exemplifies the segmentation of the riveting process into distinct stages by assigning a downward loading velocity of 0.5 mm/s to the die, enabling a systematic demonstration of process dynamics across sequential phases. The termination criterion for the riveting loading is defined as a downward displacement of 2 mm. Once this displacement is reached, the pressure is maintained for 1 s. Subsequently, the unloading process commences.

To comprehensively analyze plastic deformation characteristics induced by the strain rate across distinct stages, the riveting process was partitioned into five segments [26]. The simulation process is shown in Figure 4.

The free upsetting stage: The rivet shank’s free upsetting process occurs when the rivet material fills the gap between it and the rivet hole from about 0 s to 0.9 s.

The hole wall rivet bar interference stage: From about 0.9 s to 1.5 s, the contact between the rivet shank and the hole wall occurs until interference.

The rivet-driven head formation stage: From about 1.5 s to 4 s, the rivet-driven head is gradually formed by the continuous pressure of the die, and the end of the rivet shank material is restricted by the rivet hole to flow in the radial direction, and locally restricted upsetting deformation occurs.

The holding pressure stage: At 4–5 s, the simulation enters the riveting holding-pressure stage. At this time, the die reaches the termination condition of riveting (axial displacement of 2 mm). It remains in the current position for 1 s to ensure the stability and reliability of riveting.

The unloading stage: At 5–6 s, the die unloading is simulated, and the rivet-driven head undergoes a certain elastic recovery and partial stress release process.

3.2. Model Validation

The FE model requires validation to verify the predictive accuracy of riveting parameters’ effects on rivet formation quality. This study selected two validation metrics—driven head geometry and interference fit—for model verification through a comparative analysis of simulation outputs and experimental data. Following standard industry practices, experimental interference quantification employed the longitudinal sectioning method, focusing on rivet hole diameter assessment.

To ensure the clarity and repeatability of the interference measurement, a detailed experimental procedure was strictly followed, as summarized in the flowchart of Figure 5. The specific steps are outlined below:

(1)

Sample sectioning: The riveted specimens were carefully sectioned longitudinally through the rivet hole axis using wire electrical discharge machining (WEDM) to ensure a precise and smooth cut with an accuracy of ±0.01 mm, thereby minimizing the impact on the measurement of the hole diameter.

(2)

Point definition: Along the direction of the plate thickness, a total of 8 measurement points were defined from the top (rivet-driven head side) to the bottom (countersunk head side), spaced at 0.4 mm intervals. These points were labeled A through H, as illustrated in the schematic within Figure 5.

(3)

Diameter measurement: The diameters at each predefined point (A–H) on the deformed rivet shank were measured using an OLYMPUS-DSX1000 digital microscope (Olympus, Hachioji, Tokyo) at 400× magnification. This high magnification ensured measurement accuracy.

(4)

Interference calculation: The relative interference (${\delta }_i$) at each point was calculated using Equation (6), where $D_i$ is the measured diameter at point $i$, and $D_0$ is the nominal hole diameter of 2.6 mm.

(5)

Data validation: To ensure reliability, three riveted specimens were measured under each condition. The measurement at each point was repeated three times, and the average value was taken. Data sets with a deviation exceeding 5% were discarded and remeasured.

Using the OLYMPUS-DSX1000 digital microscope, the rivet-driven head size and the diameter of the rivet shank after riveting were measured. To investigate the impact of varying feed rates on the quality of rivet-driven heads, numerical simulation validation was performed at feed rates of 1, 5, and 10 mm/s within the range of 1–10 mm/s. The size of the rivet-driven heads and the diameter of the rivet shanks after expansion were, respectively, measured for three riveted parts under different feed rates. Then, the value was compared and analyzed with the numerical simulation results. The rivet-driven head dimensions of the nine riveted specimens measured by numerical simulation and test are shown in

Table 2. Numerical simulation results and experimental results of the rivet-driven head size.

	Upper Rivet-Driven Head Diameter/mm	Maximum Rivet-Driven Head Diameter/mm	Lower Diameter of Rivet-Driven Head/mm	Height of Rivet-Driven Head/mm
Specimen 1 of 1 mm/s	3.687	3.911	3.653	0.9823
Specimen 2 of 1 mm/s	3.642	3.901	3.594	0.995
Specimen 3 of 1 mm/s	3.627	3.851	3.619	1.01
FE model of 1 mm/s	3.555	3.936	3.62	1.031
Maximum Tolerance of 1 mm/s	3.71%	2.16%	0.91%	4.75%
Specimen 1 of 5 mm/s	3.637	3.926	3.597	1.007
Specimen 1 of 5 mm/s	3.653	3.867	3.624	0.985
Specimen 1 of 5 mm/s	3.679	3.941	3.663	0.997
FE model of 5 mm/s	3.55	3.944	3.617	1.03
Maximum Tolerance of 5 mm/s	2.45%	1.95%	1.27%	4.37%
Specimen 1 of 10 mm/s	3.647	3.937	3.657	1.021
Specimen 1 of 10 mm/s	3.69	3.879	3.614	0.989
Specimen 1 of 10 mm/s	3.661	3.926	3.594	0.983
FE model of 10 mm/s	3.56	3.947	3.622	1.034
Maximum Tolerance of 10 mm/s	3.65%	1.72%	0.96%	4.93%

below.

As Table 2 shows, the rivet-driven head size measurements obtained from the riveting tests match very well with the simulated values. The maximum errors of the tests and simulations are 3.71%, 2.16%, 1.27%, and 4.93%, with no more than 5% error for each size.

The interference value can effectively reflect the deformation condition of the rivet holes. In this study, the location for measuring the interference value is consistent with the location in the numerical simulation results of the riveting process. That is, along the direction of the plate thickness, the upper plate and the lower plate are measured, and the spacing of the measurement points is set to 0.4 mm. A total of eight measurement points were set up and numbered from A to H from top to bottom, as shown in Figure 6.

Regarding interference quantification, relative interference serves as a standardized parameter for evaluating interference fit magnitudes across varying hole diameters, as shown in Equation (6).

$${\delta }_i={\displaystyle \frac{D_i-D_0}{D_0}}$$

(6)

where i is the measurement point on the rivet shank, D_i is the diameter at each measurement point on the rivet shank after pressure riveting, D₀ is the nominal hole diameter, δ_i is the nominal hole diameter, and δ_i is the relative interference.

To quantitatively represent the uniformity of the rivet interference distribution along the plate’s thickness direction, a coefficient of standard deviation, denoted as V, is defined to describe the degree of uniformity of the relative interference values at all the measuring points [27]. The expression is given in Equation (7):

$$CV={\displaystyle \frac{\sqrt{\frac{1}{N}\sum _{i=1}^n{\left({\delta }_i-{\delta }_a\right)}^2}}{{\delta }_a}}$$

(7)

δ_a is the average relative interference value at each measurement point after pressure riveting. V is the standard deviation coefficient of the relative interference values at each measurement point. It can be concluded that the smaller the difference in V values, the better the uniformity of the rivet interference value, and, consequently, the better the riveting quality of the joint.

To ensure the reliability of the data, three riveted specimens were measured. The average interference of each measurement point was determined and compared with the simulation results, which are presented in Figure 7.

The comparison results show that the trends along the plate thickness direction are generally consistent with the interference value, regardless of whether measured in the test or simulation. When the riveting feed rates are 1, 5, and 10 mm/s, the simulated average interferences are 2.45%, 2.42%, and 2.39%, respectively, while the experimental average interferences are 2.89%, 2.85%, and 2.82% [28], and the coefficients of standard deviation obtained from the simulation and the experiment are 1.05 and 0.92, 1.07 and 0.93, and 1.08 and 0.94, respectively. A maximum relative error of 0.14 indicates a relatively high level of credibility. Through comparison, the constructed FE model can accurately simulate the actual riveting process.

Furthermore, to provide a more comprehensive validation of the dynamic riveting process, the riveting force–displacement history was recorded during both experiments and numerical simulations. A comparison for a representative case (at a constant feed rate of 5 mm/s) is presented in Figure 8. The simulated force curve agrees well with the experimental data, capturing the characteristic stages of the process: the initial low-force free upsetting stage, a sharp rise upon contact and interference with the hole wall (the hole wall rivet bar interference stage), and the final steep increase during the rivet-driven head formation stage. The peak riveting force from the simulation was 8.92 kN, compared to the experimental average of 9.45 kN, resulting in a relative error of 5.6%. This close agreement in both the trend and magnitude of the force history further reinforces the predictive accuracy and reliability of the developed finite element model.

4. Experimental Results and Discussion

Prior to experimental analysis, measurement uncertainties were systematically evaluated to ensure the reliability of the results. The automated drilling–riveting system is equipped with displacement sensors with a resolution of ±0.001 mm and load cells with an accuracy of ±0.5% of the full scale. Measurements of the driven head diameter taken with an OLYMPUS-DSX1000 digital microscope showed uncertainty of ±0.012 mm, while interference measurements had uncertainty of ±0.17%. The observed variations in driven head diameter (0.81%) and interference (2.3%) across different feed rates were substantially greater than the corresponding measurement uncertainties (0.31% and 0.17%, respectively). Repeatability tests (n = 3) yielded a coefficient of variation below 2.3%, confirming that the observed differences stem from variations in process parameters rather than measurement error.

As described earlier, the riveting process is divided into five stages: free upsetting, hole wall rivet bar interference, rivet-driven head formation, holding pressure, and unloading. Although the holding time during the pressure stage influences the formation of the rivet-driven head, it was fixed at 1 s in this experiment, as the study focuses specifically on the effect of compound feed rates on riveting quality. Based on the validated finite element model, this section systematically examines the influence of feed rate parameters on the forming quality of 120° countersunk head rivets via numerical simulation. The research objectives are threefold:

• To understand the impact of different feed rates on the quality of the rivet-driven head.
• To study the effect of the feed rate on the free upsetting stage, hole wall rivet bar interference stage, and rivet-driven head formation stage, respectively.
• To optimize the quality of riveting using the compound feed rate.

4.1. Effects of Different Feed Rates on Riveting Quality

The feed rate [29] is an important parameter in the riveting process. To study the influence of different feed rates on the quality of the rivet-driven head, the feed rate was set to 1–10 mm/s. The riveting interference of the feed rate on the A–H measurement point is shown in

Table 3. Interference at reference points with different riveting velocities (%).

Velocity	1	2	3	4	5	6	7	8	9	10
A	8.962	8.957	9.021	8.947	8.942	8.978	8.94	8.945	8.937	8.887
B	3.119	3.089	3.08	3.055	3.037	3.02	3.01	2.997	2.982	2.974
C	2.297	2.277	2.272	2.284	2.285	2.297	2.288	2.287	2.277	2.272
D	1.728	1.708	1.685	1.702	1.699	1.704	1.707	1.71	1.699	1.704
E	1.593	1.572	1.547	1.566	1.569	1.567	1.56	1.567	1.554	1.55
F	0.827	0.809	0.788	0.81	0.811	0.813	0.8	0.807	0.803	0.795
G	0.68	0.672	0.637	0.648	0.648	0.628	0.625	0.64	0.625	0.614
H	0.381	0.374	0.332	0.361	0.375	0.383	0.348	0.355	0.347	0.321
Average value	2.447	2.432	2.42	2.422	2.421	2.424	2.413	2.413	2.403	2.39

, and the different feed rates used during the riveting simulation of the rivet-driven head characteristics are shown in

Table 4. Characteristic parameters of rivet-driven head with different riveting velocities (mm).

Velocity	1	2	3	4	5	6	7	8	9	10
Upper diameter	3.555	3.544	3.544	3.552	3.55	3.55	3.549	3.559	3.558	3.56
Bottom diameter	3.618	3.619	3.62	3.62	3.617	3.621	3.62	3.621	3.622	3.622
Maximum diameter	3.936	3.94	3.942	3.943	3.944	3.944	3.946	3.946	3.947	3.947
Heights	1.031	1.034	1.035	1.034	1.03	1.033	1.036	1.035	1.032	1.034
Average diameter	3.701	3.701	3.702	3.704	3.704	3.705	3.705	3.708	3.709	3.71

.

According to the above simulation experimental data and Figure 9, it can be found that as the pressure feed rate increases, the average amount of interference shows an opposing trend. In contrast, the average diameter of the rivet-driven head shows a growing trend. The response of the rate to the amount of interference and the size of the rivet-driven head is inversely proportional, so it is necessary to study the effect of the compound feed rate on the different stages of the riveting process.

4.2. Effect of Compound Feed Rate on the Different Stages of the Riveting Process

To clarify the stage-dependent influences, the riveting process is decomposed into three key phases: free upsetting (Stage 1), hole wall interference (Stage 2), and driven head formation (Stage 3). The feed rates for each stage are denoted as v1-v2-v3 (e.g., “1-1-5” signifies v1 = 1 mm/s, v2 = 1 mm/s, and v3 = 5 mm/s). A–H represent the same original measurement points, each set to 0.4 mm. Through the control variables method, the feed rate of a certain stage is increased to observe the amount of interference and the quality of the rivet-driven head [30] after riveting is completed.

Table 5. Relative interference (%) and characteristic parameters of rivet-driven head (mm) for compound-rate riveting.

Velocity	1	5	10	1-1-5	1-1-10	1-5-1	1-10-1	5-1-1	10-1-1
A	8.962	8.942	8.887	8.965	8.798	9.019	9.023	8.989	8.882
B	3.119	3.037	2.974	3.063	3.015	3.069	3.061	3.124	3.14
C	2.297	2.285	2.272	2.29	2.29	2.289	2.281	2.29	2.323
D	1.728	1.699	1.704	1.702	1.717	1.729	1.716	1.715	1.754
E	1.593	1.569	1.55	1.566	1.563	1.603	1.592	1.582	1.623
F	0.827	0.811	0.795	0.803	0.802	0.837	0.827	0.811	0.829
G	0.68	0.648	0.614	0.655	0.644	0.685	0.671	0.68	0.692
H	0.381	0.375	0.321	0.332	0.305	0.405	0.393	0.378	0.428
Average interference	2.448	2.421	2.39	2.422	2.392	2.455	2.446	2.446	2.459
Upper diameter	3.555	3.55	3.56	3.556	3.567	3.542	3.543	3.556	3.543
Bottom diameter	3.617	3.617	3.622	3.622	3.626	3.616	3.615	3.62	3.621
Maximum diameter	3.936	3.944	3.947	3.941	3.944	3.937	3.937	3.94	3.937
Average diameter	3.703	3.704	3.71	3.706	3.713	3.698	3.698	3.706	3.701

lists the relative amount of interference at each reference point, as well as the upper end diameter, lower end diameter, and maximum diameter of the rivet-driven head after the rate changes in different phases. To address the statistical significance of the observed differences, the variability in key quality metrics was quantified via standard deviation analysis. While the standard deviation coefficient (V) of relative interference serves as one of the four primary optimization objectives in our multi-objective framework, additional statistical measures were employed to validate the consistency of other parameters. For rivet-driven head dimensions, the standard deviations across triplicate measurements remained below 2.3% of the mean values, confirming that the observed variations in diameter (0.81%) and height across different feed rates significantly exceed measurement uncertainty. Similarly, for interference values, the experimental standard deviations at each measurement point (A–H) were consistently below 0.15%, while the feed-rate-induced variations exceeded 2.3%. This statistical evidence confirms that the differences observed between various compound feed rate schemes are indeed significant and attributable to process parameters rather than experimental variability.

As shown in Figure 10, a comparison of the three sets of data—1, 1-1-5, and 1-1-10—reveals the following insights:

(1)

Accelerating the process during the free upsetting stage increases the average diameter of the rivet-driven head while simultaneously reducing the amount of interference. The results suggest that the upsetting and forming stages primarily influence the average diameter of the rivet-driven head. Notably, there is an inverse relationship between the average diameter of the rivet-driven head and the average interference.

(2)

A comparison of the 1, 1-5-1, and 1-10-1 sets shows that the average diameter of the rivet-driven head fluctuates slightly, while the average interference decreases. According to the experimental results, the interference is mainly affected by the interference phase of the hole wall rivet bar. Increasing the rate during the interference phase of the hole wall rivet bar reduces the average interference value.

(3)

Comparing the 1, 5-1-1, and 10-1-1 sets shows that increasing the rate during the upsetting stage causes a slight decrease in the interference, followed by a rapid increase. At the same time, the diameter of the rivet-driven head first increases and then decreases.

(4)

A comparison of the 5, 5-1-1, and 10, 10-1-1 sets reveals that decelerating the feed rate during the interference and rivet-driven head formation stages leads to a significant increase in the amount of interference. In the case of 5-1-1, the average diameter of the rivet-driven head increases slightly, while in the 10-1-1 case, the diameter decreases significantly.

Since the lower end diameter of the rivet-driven head directly establishes contact with the plate, a larger diameter in this region is more conducive to improving the riveting quality. It can be observed that the diameter at the lower end of 5-1-1 increases, while that of 10-1-1 remains unchanged. Therefore, it can be inferred that reducing the feed rate at different stages contributes to enhancing the overall quality of the riveted joint. The riveting experiment is carried out at a feed rate of 10 mm per second. The rate is reduced in the free upsetting stage, hole wall interference stage, and rivet-driven head formation stage to observe the changes in the amount of interference and the rivet head diameter; the specific data are shown in

Table 6. Relative interference (%) and characteristic parameters of rivet-driven head (mm) for compound-rate riveting.

Velocity	10	1-10-10	10-1-10	10-10-1	10-1-1
A	8.887	8.858	8.822	9.028	8.882
B	2.974	2.947	3.02	3.066	3.14
C	2.272	2.262	2.292	2.276	2.323
D	1.704	1.702	1.718	1.708	1.754
E	1.55	1.552	1.568	1.572	1.623
F	0.795	0.807	0.792	0.815	0.829
G	0.614	0.62	0.642	0.662	0.692
H	0.321	0.339	0.356	0.362	0.428
Average interference	2.39	2.386	2.401	2.436	2.459
Standard deviation	2.591	2.578	2.565	2.627	2.57
Upper diameter	3.56	3.562	3.576	3.539	3.543
Bottom diameter	3.622	3.622	3.626	3.617	3.621
Maximum diameter	3.947	3.946	3.946	3.94	3.937
Average diameter	3.71	3.71	3.716	3.698	3.701

.

As shown in Figure 11 and Figure 12, compared to a constant feed rate of 10 mm/s, reducing the feed rate during the free upsetting stage alone has a negligible effect. However, reducing the feed rate individually or simultaneously during the hole wall interference stage and the driven head formation stage tends to increase the interference value. Additionally, when the rate is reduced solely during the hole wall rivet bar interference stage, there is an observable tendency for the diameter of the rivet-driven head to increase.

4.3. Results of the Multi-Objective Integrated Optimization Design

Multi-objective optimization was conducted based on the riveting quality criteria defined in Section 2.2, aiming to balance the key performance indicators, including the interference value, its uniformity, and the rivet-driven head geometry.

The design variables are the feed rates at different stages, and the obtained factors and levels are shown in

Table 7. Design variables and levels.

Design Variable	Description	Level 1	Level 2	Level 3
X1	Free upsetting stage/mm	1	5	10
X2	Hole wall rivet bar interference stage/mm	1	5	10
X3	Rivet-driven head formation stage/mm	1	5	10

.

The functional expression of the established multi-objective optimization model is shown in Equation (8):

$$\begin{array}{c}\mathrm{find}x={\left(x_1,x_2,x_3\right)}^T\\ \left\{\begin{array}{c}\max \left\{{\delta }_a\left(x\right),d_a\left(x\right),d_b\left(x\right),-V\left(x\right)\right\}\\ {\delta }_A\left(x\right)\to 3.0\\ {\delta }_H\left(x\right)\to 3.0\end{array}\right.\end{array}$$

(8)

δ_a is the average amount of interference; d_a is the average diameter of the rivet-driven head; d_b is the bottom diameter of the rivet-driven head; V is the coefficient of standard deviation; δ_A is the interference value at point A; δ_H is the interference value at point H.

The results of the orthogonal experimental design with three factors and three levels are shown in

Table 8. Orthogonal experimental design with three factors and three levels, as well as simulation results.

Experiment No.	X1	X2	X3	δa	δA	δH	V	da	db
1	1	1	1	2.448	8.962	0.381	2.602	3.703	3.619
2	1	1	5	2.422	8.965	0.332	2.613	3.706	3.622
3	1	1	10	2.392	8.798	0.305	2.562	3.713	3.626
4	1	5	1	2.455	9.019	0.405	2.615	3.698	3.616
5	1	5	5	2.437	8.983	0.393	2.607	3.705	3.618
6	1	5	10	2.398	8.808	0.363	2.556	3.711	3.624
7	1	10	1	2.446	9.023	0.393	2.620	3.698	3.615
8	1	10	5	2.418	9.085	0.324	2.655	3.706	3.617
9	1	10	10	2.386	8.858	0.339	2.578	3.710	3.622
10	5	1	1	2.446	8.989	0.378	2.614	3.706	3.620
11	5	1	5	2.417	8.946	0.338	2.609	3.708	3.621
12	5	1	10	2.386	8.767	0.330	2.552	3.713	3.625
13	5	5	1	2.450	9.033	0.402	2.622	3.700	3.617
14	5	5	5	2.421	8.942	0.375	2.600	3.704	3.617
15	5	5	10	2.414	8.832	0.403	2.558	3.713	3.625
16	5	10	1	2.444	9.008	0.398	2.614	3.701	3.617
17	5	10	5	2.388	9.162	0.321	2.677	3.704	3.618
18	5	10	10	2.410	8.933	0.357	2.598	3.712	3.624
19	10	1	1	2.459	8.882	0.428	2.570	3.700	3.621
20	10	1	5	2.414	8.898	0.360	2.592	3.706	3.622
21	10	1	10	2.401	8.822	0.356	2.565	3.716	3.626
22	10	5	1	2.469	9.047	0.457	2.616	3.701	3.619
23	10	5	5	2.424	8.941	0.383	2.599	3.704	3.620
24	10	5	10	2.381	8.734	0.348	2.539	3.713	3.623
25	10	10	1	2.436	9.028	0.362	2.628	3.699	3.617
26	10	10	5	2.429	9.102	0.338	2.658	3.704	3.618
27	10	10	10	2.390	8.887	0.321	2.591	3.710	3.622

.

The results of the “grey relational generation” are shown in

Table 9. Gray relational analysis results.

Reference Experiment No.	Gray Relational Generation						Gray relational Coefficient						Gray Relational Grade
	δa	δA	δH	V	da	db	δa	δA	δH	V	da	db
	1	1	1	1	1	1	1	1	1	1	1	1	1
1	0.677	0.341	0.340	0.521	0.410	0.425	0.144	0.041	0.057	0.054	0.094	0.071	0.460
2	0.482	0.341	0.336	0.483	0.477	0.593	0.102	0.041	0.057	0.050	0.110	0.098	0.458
3	0.362	0.347	0.333	0.749	0.699	0.960	0.077	0.042	0.056	0.077	0.161	0.159	0.572
4	0.748	0.339	0.342	0.476	0.333	0.359	0.159	0.041	0.058	0.049	0.077	0.060	0.442
5	0.575	0.340	0.341	0.503	0.435	0.412	0.122	0.041	0.058	0.052	0.100	0.068	0.440
6	0.380	0.347	0.338	0.806	0.623	0.698	0.081	0.042	0.057	0.083	0.143	0.116	0.522
7	0.648	0.338	0.341	0.460	0.335	0.333	0.137	0.041	0.058	0.047	0.077	0.055	0.415
8	0.460	0.336	0.335	0.373	0.454	0.386	0.098	0.040	0.057	0.038	0.104	0.064	0.401
9	0.344	0.345	0.336	0.637	0.595	0.592	0.073	0.041	0.057	0.066	0.137	0.098	0.472
10	0.654	0.340	0.339	0.480	0.455	0.481	0.139	0.041	0.057	0.049	0.105	0.080	0.471
11	0.455	0.341	0.336	0.495	0.521	0.528	0.096	0.041	0.057	0.051	0.120	0.088	0.453
12	0.346	0.348	0.335	0.841	0.752	0.834	0.073	0.042	0.057	0.087	0.173	0.138	0.570
13	0.696	0.338	0.342	0.454	0.356	0.382	0.148	0.041	0.058	0.047	0.082	0.063	0.438
14	0.474	0.341	0.339	0.530	0.415	0.390	0.101	0.041	0.057	0.055	0.096	0.065	0.414
15	0.443	0.346	0.342	0.789	0.726	0.747	0.094	0.041	0.058	0.081	0.167	0.124	0.565
16	0.635	0.339	0.341	0.478	0.364	0.379	0.135	0.041	0.058	0.049	0.084	0.063	0.429
17	0.352	0.333	0.335	0.333	0.431	0.412	0.075	0.040	0.057	0.034	0.099	0.068	0.373
18	0.426	0.342	0.338	0.538	0.686	0.691	0.090	0.041	0.057	0.055	0.158	0.115	0.516
19	0.807	0.344	0.344	0.690	0.362	0.527	0.171	0.041	0.058	0.071	0.083	0.088	0.512
20	0.441	0.343	0.338	0.568	0.457	0.581	0.094	0.041	0.057	0.058	0.105	0.096	0.452
21	0.393	0.346	0.338	0.725	1.000	1.000	0.083	0.042	0.057	0.075	0.230	0.166	0.652
22	1.000	0.338	0.346	0.472	0.373	0.440	0.212	0.041	0.059	0.049	0.086	0.073	0.518
23	0.493	0.342	0.340	0.537	0.427	0.457	0.104	0.041	0.057	0.055	0.098	0.076	0.432
24	0.333	0.350	0.337	1.000	0.730	0.641	0.071	0.042	0.057	0.103	0.168	0.106	0.547
25	0.571	0.338	0.338	0.438	0.342	0.393	0.121	0.041	0.057	0.045	0.079	0.065	0.408
26	0.520	0.336	0.336	0.367	0.421	0.402	0.110	0.040	0.057	0.038	0.097	0.067	0.409
27	0.356	0.344	0.335	0.568	0.573	0.572	0.075	0.041	0.057	0.059	0.132	0.095	0.459

. The entropy weighting method was used to assign weights to each performance indicator. The weight values of δ_a, δ_A, δ_H, V, d_a, and d_b were 0.212, 0.12, 0.169, 0.103, 0.23, and 0.166, respectively. The gray relational grades of each experimental group were calculated, as shown in Table 8.

As shown in Table 9, the 21st experimental group achieved the highest gray relational grade value of 0.652, indicating that the compound feed rate scheme of 10-1-10 mm/s yields the best overall riveting quality against the defined optimization criteria. Under this scheme, the average interference increased by 0.46%, while the interference at point A decreased by 0.73%. The standard deviation of interference was reduced by 1%, reflecting improved uniformity. In addition, the average and lower-end diameters of the rivet-driven head increased by 0.16% and 0.11%, respectively, and the interference at point H rose significantly by 10.9%. These improvements align consistently with the multi-objective optimization targets.

5. Conclusions

This study investigates the influence of compound feed rates on the deformation mechanism and quality of HB6316 countersunk head rivets during riveting. Based on the plastic deformation behavior, the riveting process is divided into three stages: free upsetting, hole wall rivet bar interference, and rivet-driven head formation. A three-dimensional finite element model was developed to simulate the riveting process and validated experimentally by comparing predicted and measured rivet-driven head dimensions, interference distributions, and load-displacement responses. Using the validated model, the effects of compound feed rates on the riveting deformation mechanism and driven head quality were systematically analyzed. The results demonstrate that compound feed rates significantly affect the forming quality of HB6316 countersunk rivets, and an optimal scheme was identified through multi-objective integrated optimization. The main conclusions are as follows:

(1)

When the riveting feed rate increases, it leads to a decrease in the average interference, while the average diameter of the rivet-driven head increases at the same time.

(2)

Increasing the feed rate during the upsetting and hole wall interference stages reduces the interference value. In contrast, increasing the feed rate during the driven head formation stage increases the driven head diameter. This confirms that the feed rate in different stages has distinct effects on the final rivet geometry and interference.

(3)

Reducing the rate either separately or simultaneously during the interference stage between the hole wall rivet bar and the rivet-driven head formation stage tends to increase the interference value. Additionally, when the rate is reduced solely during the hole wall rivet bar interference stage, there is an observable tendency for the diameter of the rivet-driven head to increase.

(4)

By integrating gray relational analysis with the entropy weighting method and referring to the established riveting optimization criteria, it is concluded that setting the compound speed to 10-1-10 achieves the optimal outcome for riveting quality. After optimization, the average interference increases by 0.46%, while the interference at point A decreases by 0.73%. The standard deviation of the interference drops by 1%, indicating a more consistent riveting quality. Additionally, the average diameter of the rivet-driven head and the diameter of the lower-end driven head increase by 0.16% and 0.11% respectively, and the interference at point H rises by 10.9%. Overall, these adjustments to the compound speed effectively optimize riveting quality.

While the proposed methodology of decomposing the riveting process and optimizing stage-dependent feed rates is universally applicable in principle, the specific optimal parameters identified in this study (e.g., 10-1-10 mm/s) are tailored to the 2A10-2A12 aluminum alloy combination. The deformation mechanisms and material flow are highly dependent on the material’s constitutive behavior, particularly the yield strength and strain rate sensitivity. For instance, applying these parameters to high-strength steels (e.g., 30CrMnSiA, with a yield strength of about 1100 MPa, significantly higher than about 320 MPa for 2A12) would likely require different feed rates and higher riveting forces to achieve adequate plastic deformation, necessitating a separate finite element model and optimization study for such material systems.

Future investigations will incorporate microhardness measurements and microstructural analysis to build upon the present findings and provide a more comprehensive understanding at the microscale.