1. Introduction
In modern engineering applications, various techniques are employed to enhance the durability of components under pressure and increase safety levels. One of these techniques is autofrettage, a crucial manufacturing process that involves subjecting a cylinder to high internal pressure, inducing plastic deformation. Upon pressure release, the cylinder’s material attempts to revert to its original shape, resulting in beneficial residual compressive stresses within and residual tensile stresses outside the cylinder. These residual stresses enhance the cylinder’s ability to withstand higher pressures by reducing material stress. Hydraulic autofrettage, utilizing fluid pressure, and mechanical autofrettage, employing mandrels, are the two primary methods used to enhance the pressure-bearing capacity and fatigue life of cylinders. Widely adopted across industries like aerospace, automotive, and pressure vessel manufacturing, autofrettage significantly improves the durability and safety of high-pressure components [
1,
2,
3,
4,
5,
6,
7].
The most significant factors that can affect the residual stress field obtained from hydraulic autofrettage are autofrettage pressure, wall thickness, and material properties [
8]. The variability of these factors can create a high level of variability in the residual stress response. Therefore, knowing the sensitivity of the residual stress response obtained from the autofrettage process to these factors is crucial in the design of pressure vessels. Fielder et al. utilized the complex variable finite element method to compute the sensitivity of the residual stress field obtained from autofrettage applied to a thick-walled spherical pressure vessel to material parameters and autofrettage pressure. According to the results obtained, it was observed that the most influential parameter on residual stresses is the autofrettage pressure, while the yield stress stands out as the most influential among material parameters [
8].
In the production process of a component resistant to high pressure, machining is often required after the autofrettage process. Machining is employed to achieve specific manufacturing tolerances in components such as the piston–cylinder pair of a high-pressure pump. Additionally, it can be applied to the outer surface of a firearm barrel to lighten the barrel in regions where the internal pressure of the barrel is low or to create grooves on the inner bore to impart a rotational motion to the bullet. As a result of these processes, some of the beneficial residual stresses created by the autofrettage process are lost, and a new residual stress distribution is formed in the cylinder wall. This affects the load-bearing capacity of the component during service. Therefore, determining the residual stress loss caused by the machining process is crucial for accurately estimating the pressure-carrying capacity of the designed cylinder [
9,
10].
Parker et al. examined the residual stress fields resulting from internal and external material removal operations applied to a thick-walled autofrettaged cylinder for materials exhibiting both the Bauschinger effect and those that do not show it [
11]. Perl et al. developed an analytical model for calculating the new residual stress field resulting from the internal, external, and combined processing of a firearm barrel after autofrettage. The model treats partially autofrettaged and fully autofrettaged barrels separately. Additionally, in this study, they validated the analytical model developed with finite element analysis [
12]. Jahed et al. investigated the effect of material removal on residual stress using an analytical method that accounts for the material’s true unloading behavior [
13]. Hameed et al. examined the effect of internal and external machining on residual stress using the finite element method. The results indicated that the maximum circumferential stress was more sensitive to internal machining than external machining and that material removal did not alter the elastic–plastic interface [
14]. Bhatnagar et al. analyzed the effect of internal and external machining applied to a compound tube after autofrettage on residual stress loss analytically and numerically. The analysis revealed that the sequence of operations led to different results in residual stress loss when both internal and external machining were applied together [
15].
Despite the extensive research on autofrettage and its effects on residual stress, there is a notable gap in the literature regarding the quantitative impact of material parameters on the safety factor of autofrettaged cylinders, especially when different machining processes are applied post-autofrettage. This study advances the field by providing a detailed analysis of how yield strength, Young’s modulus, and tangent modulus influence the safety factor under various scenarios. By integrating the Taguchi experimental design, finite element simulations, ANOVA, and regression analysis, this research offers a comprehensive approach to understanding and predicting the behavior of autofrettaged cylinders. The novel contribution lies in quantifying the effects of these material parameters and validating the findings with both numerical and analytical methods, filling a critical gap in the existing research.
In this study, the effect of material parameters such as yield strength, Young’s modulus, and tangent modulus on the safety factor for the working pressure of autofrettaged cylinders has been investigated for the following three scenarios: with no machining, only internal machining, and only external machining. The working pressure was chosen as 400 MPa, which is a pressure value commonly experienced in pressure vessels and artillery guns [
9,
16,
17]. Initially, a series of finite element (FE) simulations were conducted based on the Taguchi experimental design, where three levels were set for each of the three selected material parameters, creating a standard orthogonal array
L9. The machining simulation after autofrettage simulation was performed using the element birth and death method available in the ANSYS 2023 R1. The results of nine FE simulations, conducted according to the Taguchi experimental design, were transformed into signal-to-noise (S/N) ratios to determine the best settings. Analysis of variance (ANOVA) was used to understand how much each factor contributed and how significant they were. To ensure the accuracy of the FE model, its outcomes were compared with an analytical method based on the deformation theory of plasticity, showing close agreement. Finally, a regression analysis predicted responses for different parameter values, with confirmation simulations confirming the accuracy of the regression model. The analysis indicated that among autofrettaged cylinders across three scenarios—without machining, with only internal machining, and with only external machining—the parameter most significantly affecting the safety factor is
Sy. Hameed et al. also highlighted the crucial role of yield strength in residual strain outcomes, noting that materials with higher yield strength exhibit better performance in terms of residual strain upon pressure removal, which aligns with our findings [
14]. In all three scenarios, the influence of
E and
T on the safety factor is relatively minimal. However,
T exhibits a greater impact than
E. As
E decreases and
T increases, there is a slight tendency for the safety factor to increase. A strong agreement is noted between the numerical and analytical findings. Generally, the safety factor tends to show lower values in the analytical approach compared to the numerical approach across all scenarios. The numerical analysis results suggest that internal machining results in a higher loss of residual stress compared to external machining.
3. Results and Discussion
3.1. Taguchi Analysis
In this study, the safety factor for three different scenarios of an autofrettaged cylinder was investigated: without machining, with only internal machining at a thickness of 5 mm, and with only external machining at a thickness of 5 mm. The effect of the factors listed in
Table 1 was examined using the Taguchi
L9 (3
3) orthogonal array design. The experimental matrix created using the Minitab 19 statistical software is presented in
Table 2. The safety factor for the operating pressure of autofrettaged cylinders without machining, with only internal machining, and with only external machining was taken as the response parameter. Since the goal is to maximize each response parameter, a type of S/N ratio where larger is better was considered.
FE Analyses were conducted for each set of parameters specified in
Table 2 of the experimental matrix. Subsequently, the calculated simulation responses were transformed into signal-to-noise (S/N) ratios using the Minitab 19 software. Regardless of the objective function, a larger S/N ratio corresponds to better performance characteristics. The calculated signal-to-noise (S/N) ratio values (mean) for the safety factor of the autofrettaged cylinder without machining are provided in
Table 3 and graphically presented in
Figure 4.
The analysis of
Figure 5 reveals that an increase in the yield strength of the material undergoing the autofrettage process without machining can enhance the S/N ratio by 4 to 5 times, irrespective of the Young’s modulus value. Conversely, at a tangent modulus value of 0, the S/N ratio rises proportionally with the increase in Young’s modulus. Notably, the optimal S/N ratios are observed with the maximum levels of yield strength and tangent modulus within the examined range, along with the lowest value of Young’s modulus.
The maximum effectiveness ranks were calculated based on delta values representing the variability between the highest and lowest average response values for each factor. According to the delta values in
Table 3, the most influential parameter on the response for the autofrettaged cylinder with only internal machining is
Sy. It is clearly seen from the graphical data (
Figure 4) that the average S/N ratio increases as
Sy and
T increase. This indicates that the safety factor of the autofrettaged cylinder without machining increases as
Sy and
T increase. However, the safety factor decreases as
E values increase. The dashed line in
Figure 4 represents the overall average of all S/N ratios. The calculated S/N ratio values (average) for the safety factor of the autofrettaged cylinder with only internal machining applied are provided in
Table 4 and presented graphically in
Figure 6.
A similar outcome to the autofrettage process without machining is evident in the case of the process involving internal or external machining, as depicted in
Figure 7 and
Figure 8. The distinction between them becomes apparent at the maximum value that the S/N ratio can attain.
According to the delta values in
Table 4, the most influential parameter on the response for the autofrettaged cylinder with only internal machining is
Sy. It is clearly seen from the graphical data (
Figure 6) that the average S/N ratio increases as
Sy and
T increase. This means that the safety factor of the autofrettaged cylinder with only internal machining increases as
Sy and
T increase. However, the safety factor decreases as
E values increase. The calculated S/N ratio values (average) for the safety factor of the autofrettaged cylinder with only external machining are provided in
Table 5 and presented graphically in
Figure 9.
According to the Delta values in
Table 5, the most influential parameter on the response for the autofrettaged cylinder with only external machining is
Sy. As observed from the graphical data (
Figure 9), the average S/N ratio increases with the increase in
Sy and
T. This implies that the safety factor of the autofrettaged cylinder with only external machining increases with increasing
Sy and
T. However, the safety factor decreases with increasing
E values.
According to the analysis results obtained using Taguchi’s robust process design, in order to achieve the maximum safety factor in all three scenarios, the values of Sy, E, and T should be 0.7 GPa, 100 GPa, and 20 GPa, respectively.
3.2. Analysis of Variance
The Taguchi methodology does not directly provide the impact of process parameters on specific outputs. Therefore, statistical tools are especially required to calculate the relative importance and effects of each process parameter on specific outputs. The effectiveness of each parameter was evaluated using analysis of variance (ANOVA). This method can clearly identify how much each process parameter influences the response and determine the contribution of each parameter. The ANOVA analysis was conducted at a 95% confidence level. The importance of process parameters was determined by comparing the
p-values. The lower the
p-value, the greater the effect of that parameter on the response. Generally, if the
p-value is less than 0.05, it indicates that the parameter has a significant effect on the process response [
24]. Furthermore, it is necessary to verify whether assumptions such as independence and homogeneity of variance are met for ANOVA [
25,
26,
27]. In this context, the Shapiro–Wilk test was used to check the normality of the samples for the analyses conducted in this study.
Table 6 shows the ANOVA results for the safety factor of the autofrettaged cylinder without machining. When examining the ANOVA results, it is observed that the
p-values for the
Sy and
T parameters are less than 0.05. Therefore, since the
p-value is less than 0.05 at a 95% confidence level, it is concluded that the factors are statistically significant in their effect on the response. Additionally, the contribution of each material parameter was calculated by dividing each individual sum of squares by the total sum of squares. The contributions of the
Sy,
E, and
T parameters to the response are 98.20%, 0.41%, and 1.29%, respectively. Yield strength is the most influential factor on the safety factor of the machined autofrettaged cylinder, while the effects of
E and
T are very low.
Table 7 displays the ANOVA results for the safety factor of the autofrettaged cylinder with only internal machining. Upon examining the ANOVA results, it is observed that the
p-values for all material parameters are less than 0.05. Therefore, since the
p-value is less than 0.05 at a 95% confidence level, the factors are statistically significant in their effect on the response. The contributions of the
Sy,
E, and
T parameters to the response are 98.20%, 0.4%, and 1.26%, respectively. Yield strength is the most influential factor on the safety factor of the autofrettaged cylinder with only internal machining, while the effects of the other two factors are very low.
Table 8 shows the ANOVA results for the safety factor of the autofrettaged cylinder with only external machining. Upon examining the ANOVA results, it is observed that the
p-values for the
Sy and
T parameters are less than 0.05. Therefore, since the
p-value is less than 0.05 at a 95% confidence level, it is concluded that the factors have a statistically significant effect on the response. The contributions of the
Sy and
T parameters to the response are 98.19% and 1.39%, respectively. While
Sy is the most influential factor on the safety factor of the autofrettaged cylinder with only external machining, the effect of
T is very low.
E, on the other hand, does not have any effect on the response.
3.3. Comparison of the Numerical and Analytical Methods
For the validation of the FE model, analytical calculations were performed according to the plan in
Table 2, and they were compared with the FE simulation results.
Table 9 shows the safety factors numerically and analytically calculated for the autofrettaged cylinder without machining at the operating pressure according to the plan in the experiment matrix. Upon examining the results, it is observed that there is a maximum deviation of 4.37% between the numerical and analytical predictions.
Table 10 presents the safety factors calculated numerically and analytically for the autofrettaged cylinder subjected to only internal machining, following the experimental matrix plan. Upon review, we find a maximum deviation of 4.75% between the numerical and analytical estimations.
Table 11 displays the safety factors calculated numerically and analytically for the autofrettaged cylinder subjected to only external machining, following the experimental matrix plan. Upon examination of the results, we observe a maximum deviation of 5.20% between the numerical and analytical predictions.
3.4. Effect of Machining Process on Factor of Safety
Table 12 has been created to compare the changes in the safety factor determined for the autofrettaged cylinder after the internal and external machining operations. The values comprise the results obtained from numerical analysis.
Upon examination of
Table 12, it is observed that the safety factor determined for the autofrettaged cylinder with only internal machining decreases by an average of 11.94%, whereas the safety factor for the autofrettaged cylinder with external machining decreases by an average of 4.97%. This indicates that the internal machining operation leads to a greater residual stress loss compared to the external machining operation. This effect can be attributed to the distribution of equivalent residual stresses obtained after the autofrettage process, which decrease from the inner surface towards the outer surface of the cylinder. Consequently, internal machining, which directly affects the region with the highest residual stresses, causes a more significant reduction in residual stresses than external machining.
3.5. Regression Analysis
In this study, regression analyses were also employed to investigate the relationship between dependent and independent variables. A regression analysis based on numerical data was conducted for the safety factors of the autofrettaged cylinder without machining. The following equation defines the regression model.
A regression analysis based on numerical data was applied to the safety factors of the autofrettaged cylinder with only internal machining. The equation below defines the regression model.
A regression analysis based on the numerical data was conducted for the safety factors of the autofrettaged cylinder with only external machining. The equation below defines the regression model.
Upon examining the equations derived from the regression analysis, it is observed that E is negatively signed in all equations. This indicates that E has a decreasing effect on the response. By evaluating the magnitudes of the coefficients preceding each parameter in the equations, we can determine the relative influence of each parameter on the response. Specifically, Sy has the most significant impact on the safety factor, followed by T, and finally, E. In practical terms, this means that increasing the yield strength of the material will result in the most substantial improvement in the cylinder’s ability to withstand pressure, while changes in the tangent modulus and Young’s modulus will have progressively lesser effects.
In our analysis, we also considered the potential for multicollinearity among the independent variables. Multicollinearity can lead to unreliable and unstable estimates of regression coefficients, inflating the variance and making it difficult to determine the individual effect of each predictor.
To assess and address multicollinearity, we calculated the variance inflation factor (VIF) for each independent variable. A VIF value below 5 generally indicates that multicollinearity is not a serious concern. In all three scenarios, the models without interaction terms showed high values and statistically significant p-values (below 0.05) for each parameter. Additionally, the VIF values for all parameters in these non-interaction models were 1.0, indicating no multicollinearity issues. Therefore, we chose to use the simpler models without interaction terms for clarity and consistency, ensuring robust and valid findings.
3.6. Validation of the Regression Model
The next step involves conducting validation tests to confirm the predictive capability of the equations obtained from the regression analysis [
24]. For this purpose, the safety factor values obtained from simulations conducted with inputs other than the material parameters used in the Taguchi analyses are compared with the values predicted by the regression equations. Since AISI 4340 steel is commonly used in autofrettage applications, it was utilized in this validation test [
28]. The material properties of AISI 4340 steel are provided in
Table 13 [
18].
The safety factors calculated using numerical methods and regression equations are provided in
Table 14. When comparing numerical and regression results, it is observed that there is a maximum deviation of 0.0279%. Therefore, it can be concluded that the regression model has a high predictive power.