Multi-indicator optimization of riveting joint forming quality of aluminum alloy sheets based on response surface test

2.1. Clinching forming mechanism

Instead of using rivets, the rivetless joining process uses a special convex-concave die to stamp the joining plate and then uses the plastic deformation capacity of the plate to embed the plate for joining [10]. Fig. 1 illustrates the molding mechanism. Firstly, in the positioning stage, two layers of plates are stacked between the punch and the die. The punch, the die and the center of the joint are placed coaxially. As the forming process proceeds, the plates are compressed under pressure and further stretched at the neck, and the thickness of the plates at this position gradually decreases.

Fig. 1Example of figure consisting of multiple charts

a) Positioning

b) Early stage of forming

c) Late forming

d) Demolding

2.2. Geometric modeling of clinching

The axisymmetric geometric model of riveted joint created by UG software is shown in Fig. 2(a). The punch, crimping ring and concave mould are set as rigid bodies, and the upper and lower plates are set as plastic bodies. The downward speed of the punch is set to 5 mm/s, the friction coefficient between the plates is set to 0.2, and the mesh is refined using the local mesh refinement technique. The upper and lower plates are made of 6061 aluminum alloy with a thickness of 1.5 mm and their main mechanical properties are shown in Table 1 [11]. To improve the accuracy of the finite element simulation, the Lagrange increment is used as the simulation form, and the conjugate gradient method is chosen to directly simulate the boundary conditions of the upper and lower plates in the $X$, $Y$, and $Z$ directions, while keeping the position fixed. The finite element model of the riveting mould is shown in Fig. 2(b).

Fig. 2Clinching finite element modelling: d – punch diameter; D – die diameter; H – die depth; t1, t2 – upper and lower plates

a) Left chart

b) Right chart

2.3. Geometric modeling of clinching

The quality of the riveted joint is determined by measuring and comparing the interlock value $T_u$, neck thickness value $T_n$ and base thickness value $X$ of the geometric parameters of the center section of the rivet head as shown in Fig. 3. According to literature [12], the interlocking value $T_u$ reflects the reliability of the riveted joint. the higher the $T_u$, the more reliable the riveted joint is, whereas the lower the $T_u$, the more easily the connected plates can be separated. When the neck thickness value $T_n$ is lower, the joint is more brittle. In addition, the bottom thickness value $X$ determines the tensile strength of the joint. Obviously, $T_u$, $T_n$ and $X$ are closely related.

Fig. 3Three evaluation parameters

3.1. Experimental design by response surface methodology

Response surface methodology, as an optimization method combining experimental design and mathematical modelling, has the advantages of high precision and accuracy in deriving regression equations and good predictive performance. When the number of levels of test factors is large, the response surface method can effectively reduce the number of tests and can respond to the interaction between the influencing factors [13]. In the rivetless riveting and forming process, the main variables affecting the quality of riveting and forming are the punch diameter, the depth of the concave die and the stamping speed. To test the model fit, parameter significance and misfit, as well as to analyze the variance and make model judgements, a Box-Behnken Design (BBD) response surface experimental design and analysis was used to develop the predictive model. The model was optimally solved to obtain the best test scenario.

3.2. Response surface method modeling

The parameters affecting the quality of riveted joints, i.e., neck thickness value, interlock value and bottom thickness value are used as the center point of the optimization experimental design and the combination of these three parameters is used as the optimization objective.

To establish the response surface approximation model, the relationship between the design variables and the analysis objectives must be clarified and the appropriate functional relationship must be selected. The relationship between the system response $Y$ and the design variable $x$ generally satisfies Eq. (1):

where 𝑓(𝑥) is the approximation of the unknown function, $x=\left(x_1,x_2,\dots ,x_n\right)$; 𝑛 means there are 𝑛 groups of independent variables; and 𝜉 is the total error.

The polynomial representation is expressed by Eq. (2):

where, ${\beta }_i$ is the coefficient of the basis function, ${\varphi }_i\left(x\right)$ is the basis function, $i$ is the number of dimensions, and $k$ is the number of basis functions.

The Box-Behnken Combined Design method in Design-Export software was used to determine the design of the flat-bottomed rivetless joint test. A level value was selected as the response surface test design level for the upper and lower regions, centered on the optimum value point of each test single factor. The level values and design factors are shown in Table 2.

As shown in Table 3, a table of test protocols is created after each factor and its level values are entered sequentially in the Design-Expert system. The simulation is then performed according to the test protocol. The simulation results are measured for each set of factor combinations and the simulation results are filled in sequentially into the protocol combinations. Due to certain numerical fluctuations in the simulation process, the results can be slightly different even for the same set of simulation combinations. Therefore, repeat the test for several groups in the protocol.

Significance tests were performed on the linear function, second order model and third order model. To select the appropriate model, the data from the significance test, the test of misfit term and the correlation test were compared for the three models. The second-order model was recommended for neck thickness values, and the third-order model was recommended for interlock and bottom thickness values. The test results were analyzed using Design-Expert software.

3.3. Fitting and evaluation of response surface models

Based on the analysis of the simulation results in Tables 4-6, the response fitting equations for the neck thickness values, interlocking values and bottom thickness values were derived as shown in Eqs. (3-5):

where, $A$, $B$ and $C$ represent punch diameter, die depth and stamping speed respectively. $Y_1$, $Y_2$ and $Y_3$ represent neck thickness value, interlocking value, and bottom thickness value respectively.

The above fitted regression equations were further analyzed by error statistics and the results are presented in Tables 7-9. If the value of the fitted term Pr > F of ANOVA is less than 0.05, the model is significant; if the value of the fitted term Pr > F is greater than 0.05, the model is not significant, i.e., the model is well fitted throughout the regression region of the study. The regression model adequately describes the process when the value of Adj R-squared is close to the value of Pred R-squared (RAdj2-RPred2 < 0.2) and both values are high. However, if both values are low, the process is not adequately explained, and some other important factors need to be considered. When the CV is less than 10 per cent, it indicates that the reliability and accuracy of the test is high. Precision (Adeq Precision) is the effective signal to noise ratio and if it is greater than 1, the accuracy is considered reasonable [14]. Tables 4-6 shows that the fitted regression equations comply with the above test principles, indicating their adaptability (Note: SS-Sum of Squares of Variance, DF-Degrees of Freedom, MS-Mean Square, Pr > F-No Significant Effect probability, ** highly significant, * significant).

The analysis shows that the distributions of the residuals are all close to the center straight line, indicating a good adaptation of the response surface fitting model. Figs. 4-6 show the normal probability distribution of the residuals of neck thickness values, interlocking values and bottom thickness values, the distribution of the residuals with respect to the predicted values and the distribution of the predicted values with respect to the actual values, respectively. Fig. 10 shows the response surface plots of each interaction factor for neck thickness value, interlock value and bottom thickness value. The effect of any two factors on the response surface interaction can be visualized from the above figure to determine the optimum factor level for the indicator.

Fig. 4Residual error distribution of neck thickness

a) Normal probability distribution

b) Predicted vs. actual value distribution

Fig. 5Distribution diagram of interlock value residuals

a) Normal probability distribution

b) Predicted vs. actual value distribution

Fig. 6Residual distribution of bottom thickness value

a) Normal probability distribution

b) Predicted vs. actual value distribution

The results in Fig. 7 agree with the results of the ANOVA of the regression equation $Y_1$ for the values of neck thickness. From Fig. 7(b), there is a significant interaction effect of punch diameter and stamping speed. The maximum effect of mold depth on neck thickness value is followed by stamping speed, while the minimum effect of punch diameter is observed.

Fig. 7Effect of A, B and C on neck thickness values

a)$y=f\left(A,B\right)$ response surface

b)$y=f\left(A,C\right)$ response surface

b)$y=f\left(B,C\right)$ response surface

Fig. 8Effect of A, B and C on interlocking values

a)$y=f\left(A,B\right)$ response surface

b)$y=f\left(A,C\right)$ response surface

b)$y=f\left(B,C\right)$ response surface

Fig. 8 shows that die depth and punch diameter have significant effect on the interlocking values. The most significant interaction effect is between punch diameter and die depth, followed by the interaction effect between die depth and stamping speed. There is also an interaction effect between die depth and press speed.

From Fig. 9, the interaction between die depth and stamping speed is significant and is an important variable affecting the value of bottom thickness, but the interaction between punch diameter and both die depth and stamping speed respectively is not significant.

Response surface interaction analysis shows that the concave die depth, punch speed and punch diameter are all critical dimensional parameters that affect the joint. Analysis of the results of the response surface methodology shows that the ideal dimensions of the die are a punch diameter of 5.24 mm, a die depth of 1.44 mm, a punching speed of 5.00 mm/s. For the values of neck thickness, interlocking and bottom thickness, the most obvious of the pairwise interactions of neck thickness, interlocking and bottom thickness values is the pairwise interaction of punch diameter and concave die depth.

Fig. 9Effect of A, B and C on bottom thickness values

a)$y=f\left(A,B\right)$ response surface

b)$y=f\left(A,C\right)$ response surface

b)$y=f\left(B,C\right)$ response surface

4.1. Target optimization and experimental validation

The response surface methodology was used to determine the following stamping parameters: punch diameter of 5.24 mm, die depth of 1.44 mm, stamping speed of 5.00 mm/s$.$ In addition, the model was optimized using response surface software and the optimized parameters are shown in Table 10.

Numerical simulations were performed in DEFORM-3D with the friction coefficient set to 0.2. As shown in Fig. 13, the predicted thickness of the response surface is 0.420 mm at the neck, 0.412 mm at the interlock, and 0.574 mm at the bottom. This corresponds to the corresponding relative errors of 5.96 %, 3.29 % and 1.37 %, respectively, as predicted by the response surface optimization objective. According to literature [14], the interlock value $T_u$ indicates the reliability of the riveted joint. The higher the Tu value, the more reliable the riveted joint is and vice versa. The lower the $T_n$ value, the more prone the riveted joint is to fracture.

Fig. 10Joint process parameters

4.2. Experimental verification of optimization results

Practical experiments were carried out based on the desired test parameters and the obtained experimental values were compared with the model predictions to further test the validity of the response surface method. The riveting experiments were carried out on a WDW-100 microcomputer-controlled electro-hydraulic servo composite testing machine by stacking two plates on a concave mold. The punch position was adjusted to hold the plates in place, the preload force was entered to compress the plates, and the punch speed was entered in the software to move the punch downwards. The connected plates were then pressed into the recessed die until they filled the die with the preload force. The test procedure is shown in Fig. 14. After the riveting was completed, a wire cutter was used to cut the test piece along the center of the test piece to obtain an axial section of the riveted joint. The evaluation metrics were measured using a video meter and the results are shown in Fig. 11 and Table 11.

Fig. 11Rivet joint experiment

The relative errors of the objective predictions for the response surface optimization were 13.42 % for the neck thickness, 4.23 % for the interlock value and 2.23 % for the bottom thickness.

The inverted shape produced by the plates riveted using the optimized process parameters is more complete as shown in Fig. 12. According to literature [15], the range of bottom thickness value is 20 % of the plate thickness. The optimum bottom thickness value is 0.569 mm. the smaller the value of the bottom thickness, the smaller the possibility of separation of the upper and lower plates and the higher the reliability of the connection, provided that the quality of the connection is guaranteed. By comparing the simulation error, the actual test error, and the target prediction error for response surface optimization, the neck thickness value has the largest error. This is since the neck of the aluminum alloy plate is subjected to the greatest tensile force during the clamping process and the internal metal grain movement in this area is also the most intense. These results show that the target optimization prediction model provided by response surface analysis is a reasonable, fast, and efficient method.

Fig. 12Experimentally obtained joint shape

4.3. Riveted joint strength and failure mode prediction

After the riveting is completed, the plates are lapped as shown in Fig. 13, and the riveting is implemented in the center region of the lap, and then the specimens are subjected to tensile shear experiments on the tensile equipment shown in Fig. 14(a), with a tensile shear speed of 5 mm/s. According to the joint failure mode prediction method given in reference [16], the prediction equations for the tensile shear strength of the joints are derived in this paper as follows:

where $F$ is the maximum shear force that the joint can withstand; $\beta$ is the proportionality coefficient, aluminum alloy is usually taken as 0.7; $\eta$ is the rotational compensation coefficient of the joint in the case of pulling and shearing, and it is 1.55 for 1.5 mm thick aluminum alloy plate; ${\left[\sigma \right]}_u$ is the ultimate strength of the upper plate.

Under the optimal parameters obtained according to Eq. (6), the tensile shear strength of the joint is 1.739 KN. Fig. 14(b) shows the joint damage form of shear fracture of the plate, and the experimental load-displacement curves of the plate in the process of shear are shown in Fig. 15, which indicates that the experimental value of the joint tensile shear strength is 1.8 KN and is higher than that of the theoretically calculated value, with an error of only 3.4 %. The rivetless riveted joint tensile shear experiments show that the response surface method optimized joint strength to meet the requirements and get a small range of enhancement.

Fig. 13Schematic diagram of a stretched sample

Fig. 14Tensile test

a) Tensile testing equipment

b) Riveting point neck break failure

Fig. 15Load-displacement curve