Keywords

1 Introduction

Inconel 718, a nickel-based superalloy, boasts remarkable attributes such as exceptional heat resistance [1], corrosion resistance [2], fatigue resistance [3], and fracture strength [4]. It has proven its mettle by enduring extreme environments with temperatures soaring up to 600 °C. Consequently, it has emerged as the material of choice across a spectrum of industries, most notably in aerospace [5, 6] and shipbuilding [7], where its applications hold immense promise.

In the realm of aerospace engineering, Inconel 718 has firmly established its reputation as an indispensable component. It plays a pivotal role in the manufacturing of aviation jet engines [8] and finds extensive use in various industrial gas turbines [9]. Impressively, this alloy often accounts for up to half of the total weight of these high-performance engines.

Milling is the primary method employed for the manufacture of Inconel 718; however, it poses several challenges, including severe tool wear, high thermal stress, substantial cutting forces, and chattering [10, 11]. These issues must be effectively addressed to fully harness the potential of Inconel 718 in various applications. To tackle these challenges, a comprehensive investigation into the milling mechanism of Inconel 718 is imperative. The utilization of modelling and finite element simulation techniques for analyzing the cutting forces in Inconel 718 offers a powerful solution. Through these methods, the cutting process of Inconel 718 can be accurately simulated under different cutting parameters. This approach enables the swift assessment of the influence of varying cutting conditions on critical performance metrics such as cutting forces and tool wear. Importantly, it obviates the need for costly physical experiments. By leveraging these techniques, manufacturers can determine the optimal combination of machining parameters. This optimization is instrumental in mitigating tool wear, extending the service life of cutting tools, suppressing chattering, and ultimately enhancing manufacturing efficiency. Therefore, investigating the milling mechanism of Inconel 718 through modelling and finite element simulation serves as a key strategy to overcome the challenges associated with machining this high-performance alloy.

Lee et al. [12] used a series of orthogonal milling test data at different milling speeds and feeds to derive a calculation model for the milling force of a ball-end milling cutter. And for the first time, the milling force generated during the milling performed by the ball-end milling cutter was divided into edge force and shearing force. When calculating the milling force, the blade of the ball-end milling cutter was divided into a series of oblique milling elements, and considerable experiments were carried out to verify the correctness of the proposed model. Sonawane et al. [13] established a theoretical calculation model of milling force for Inconel 718 with ball-end milling cutters. This model can be used to predict the milling force, chip geometry and instantaneous shear angle during the milling process. The relationship between the shear strength of the workpiece material and the strain, strain rate and temperature were taken into account when the theoretical model was established, and the Johnson–Cook material model was used to describe the above relationship. Kao et al. [14] decomposed the milling force generated during the milling process of the ball-end milling cutter into axial, radial and tangential forces. In addition, the authors drew the stability lobe diagram of the milling process of the ball-end milling cutter through the CUTPRO software, determined the stability conditions of the milling process of the ball-end milling cutter, and verified the reliability of the milling force model of the ball-end milling cutter through experiments. Sethupathy et al. [15] established a statistical model through the response surface method, and predicted the milling force based on tool geometric parameters and processing parameters. A second-order mathematical model for calculating the milling force of AL7075-T6 aluminum alloy with a ball-end milling cutter was developed, and the correctness of the model was verified by experiments. Zhao et al. [16] used orthogonal tests to obtain the milling force coefficients and established an empirical model of milling force. Li et al. [17] established an empirical model for the prediction of milling force and cutting temperature based on the multiple linear regression method. Li et al. [18] used ABAQUS software to analyze the effects produced by polycrystalline diamond tools for machining thin walls of SiCp/Al composite material. Jin et al. [19] established an optimized finite element model based on pendulum motion, considered the performance parameters of the workpiece, and used ABAQUS software to study the rule of change of milling force during the milling process of aerospace thin-wall component. Davoudinejad et al. [20] studied the chip formation and milling force when milling Ti–6Al–4V titanium alloy under dry and cryogenic cooling conditions by using the AdvantEdge finite element cutting software. When establishing the finite element model, the Coulomb’s friction law is used to calculate the friction stress in the milling process, and the accuracy of the model is verified by comparison with the experimental results. Mebrahitom et al. [21] and Bhopale et al. [22] simulated the milling process of Aluminum 6010 and AISI 1018 cold-rolled steel using Abaqus software, and analyzed the milling force and processing parameters. Yameogo et al. [23] used Abaqus software to establish a finite element model of milling of Ti6Al4V alloy, and proposed a new Multi-Branch (MB) theory to consider the dynamic recrystallization during milling. Rahul et al. [24] used the finite element method to analyze the residual stress of the subsurface during micro-milling Ti–6Al–4V alloy.

In summary, based on abovementioned research background, in this study, novel contributions have been introduced in this paper, encompassing milling force modelling, finite element analysis, the influence of machining parameters, and practical applications. Collectively, these contributions serve to advance the understanding of ball-end milling processes and their optimization, especially for challenging materials like Inconel 718.

2 Modelling of Milling Force of Ball-End Milling Cutter

A well-developed model of milling force of ball-end milling cutter allows for a systematic analysis of how milling forces change under different conditions, such as varying cutting speeds, feed rates, and depths of cut. This information is crucial for optimizing machining parameters, leading to increased efficiency and reduced tool wear.

2.1 Establishment of Milling Force Model of Ball-End Milling Cutter

According to the theory proposed by Lee et al. [12], if the tip radius of the ball-end milling cutter could be neglected, the microelement of cutting force generated by the ball-end milling cutter during machining can be presented as:

$$\left\{ \begin{gathered} dF_{t,j} \left( {\phi_{j} \left( z \right)} \right) = K_{te} dS\left( {\phi_{j} \left( z \right)} \right) + K_{tc} h_{j} \left( {\phi_{j} \left( z \right)} \right)db \hfill \\ dF_{r,j} \left( {\phi_{j} \left( z \right)} \right) = K_{re} dS\left( {\phi_{j} \left( z \right)} \right) + K_{rc} h_{j} \left( {\phi_{j} \left( z \right)} \right)db \hfill \\ dF_{a,j} \left( {\phi_{j} \left( z \right)} \right) = K_{ae} dS\left( {\phi_{j} \left( z \right)} \right) + K_{ac} h_{j} \left( {\phi_{j} \left( z \right)} \right)db \hfill \\ \end{gathered} \right.$$
(1)

In Eq. (1), the parameters \(K_{tc}\), \(K_{rc}\), \(K_{ac}\) are named as shearing force coefficients, and parameters \(K_{te}\), \(K_{re}\), \(K_{ae}\) are called edge force coefficients. According to method outlined by Kao et al. [14], the edge and shearing force coefficients can be calculated by using equation given below:

$$\left\{ {\begin{array}{*{20}l} {K_{ac} = \frac{{\overline{F}_{zc} \left( {C_{3} C_{7} C_{15} - C_{1} C_{9} C_{15} } \right) - C_{7} C_{13} C_{15} \overline{F}_{fc} + C_{1} C_{13} C_{15} \overline{F}_{nc} }}{{C_{3} C_{7} C^{2}_{15} - C_{5} C_{7} C_{13} C_{15} - C_{1} C_{9} C^{2}_{15} + C_{1} C_{11} C_{13} C_{15} }}} \hfill \\ {K_{ae} = \frac{{\overline{F}_{ze} \left( {C_{4} C_{8} C_{16} - C_{2} C_{10} C_{16} } \right) - C_{14} C_{8} C_{16} \overline{F}_{fe} + C_{2} C_{14} C_{16} \overline{F}_{ne} }}{{C_{4} C_{8} C^{2}_{16} - C_{6} C_{8} C_{14} C_{16} - C_{2} C_{10} C^{2}_{16} + C_{2} C_{12} C_{14} C_{16} }}} \hfill \\ {K_{rc} = \frac{{C_{7} C_{15} \overline{F}_{fc} - C_{1} C_{15} \overline{F}_{nc} + \left( {C_{1} C_{11} - C_{5} C_{7} } \right)\overline{F}_{zc} }}{{C_{3} C_{7} C_{15} - C_{5} C_{7} C_{13} - C_{1} C_{9} C_{15} + C_{1} C_{11} C_{13} }}} \hfill \\ {K_{re} = \frac{{C_{8} C_{16} \overline{F}_{fe} - C_{2} C_{16} \overline{F}_{ne} + \left( {C_{2} C_{12} - C_{8} C_{6} } \right)\overline{F}_{ze} }}{{C_{4} C_{8} C_{16} - C_{6} C_{8} C_{14} - C_{2} C_{10} C_{16} + C_{2} C_{12} C_{14} }}} \hfill \\ {K_{tc} = \frac{{\overline{F}_{fc} - C_{3} K_{rc} - C_{5} K_{zc} }}{{C_{1} }}} \hfill \\ {K_{te} = \frac{{\overline{F}_{fe} - C_{4} K_{re} - C_{6} K_{ze} }}{{C_{2} }}} \hfill \\ \end{array} } \right.$$
(2)

Finally, the average milling force per rotation of the ball-end milling cutter can be calculated using the Eq. (3). The parameters C1–C16 in Eq. (3) are explained in Ref. [14].

$$\left\{ \begin{gathered} \overline{F}_{f} = \left( {C_{1} K_{tc} + C_{3} K_{rc} + C_{5} K_{ac} } \right)f_{t} + C_{2} K_{te} + C_{4} K_{re} + C_{6} K_{ae} \hfill \\ \overline{F}_{n} = \left( {C_{7} K_{tc} + C_{9} K_{rc} + C_{11} K_{ac} } \right)f_{t} + C_{8} K_{te} + C_{10} K_{re} + C_{12} K_{ae} \hfill \\ \overline{F}_{z} = \left( {C_{13} K_{rc} + C_{15} K_{ac} } \right)f_{t} + C_{14} K_{re} + C_{16} K_{ae} \hfill \\ \end{gathered} \right.$$
(3)

2.2 Influence of Machining Parameters on Milling Force

The average milling force can be computed by employing Eq. (3), and simultaneously adjusting the machining parameters will result in changes to the milling force, thereby revealing the influence and effect of machining parameters on milling force is crucial.

The feed per tooth plays a pivotal role in determining both the magnitude of the milling force and the quality of the machined surface. Consequently, to investigate the influence of the feed per tooth on the milling forces, the different feed rate per tooth is set separately as \(f_{z}\) = 0.1 mm/z, 0.2 mm/z, 0.3 mm/z, 0.4 mm/z, while keeping the radial depth of cut and axial depth of cut unchanged, namely \(a_{e} = 3\;{\text{nm}}\), \(a_{p} = 0.5\;{\text{mm}}\). In addition to feed per tooth, the tool helix angle is another important parameter that affects the milling process. Change in helix angle in ball-end milling cutter affects the cutting angle and feed angle, which in turn affects the contact between tool and workpiece. The tool helix angle is set separately as β = 15°, β = 25°, β = 35°, β = 45° in the case of ensuring other parameters remain unchanged. The results of above studies are shown in Figs. 1, 2 and 3. When the feed per tooth of the ball-end milling cutter increases, there is a corresponding upward trend observed in the milling force for each axis. Similarly, an increase in the helix angle of the ball-end milling cutter results in a proportional rise in the cutting force. This can be attributed to the expansion of the contact area between the tool and the workpiece, leading to an overall mounting trend in cutting force along all axes. Due to the significant impact of the ball-end milling cutter’s helix angle on the generated milling force in each axis, careful consideration and selection of the appropriate tool helix angle are imperative to meet specific machining requirements.

Fig. 1
2 profiles of milling force of X axis versus rotating angle of ball end milling cutter. a. 4 sine functions for f z = 0.1, 0.2, 0.3, and 0.4 millimeters. The functions are plotted between negative 300 and 2250 with varying amplitudes. b. 4 zig zag curves for beta are plotted between negative 400 and 700.

Influence of processing parameters on milling force of X-Axis: a Influence of feed per tooth; b Influence of tool helix angle

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Fig. 2
2 profiles of milling force of Y axis versus rotating angle of ball end milling cutter. a. 4 sine functions for f z = 0.1, 0.2, 0.3, and 0.4 millimeters are plotted between negative 2000 and 500 with varying amplitudes. b. 4 sine functions for beta are plotted between negative 750 and 100.

Influence of processing parameters on milling force of Y-Axis: a Influence of feed per tooth; b Influence of tool helix angle

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Fig. 3
2 profiles of milling force of Z axis versus rotating angle of ball end milling cutter. a. 4 distribution functions for f z = 0.1, 0.2, 0.3, and 0.4 millimeters are plotted between negative 300 and 950 with varying amplitudes. b. 4 functions for beta are plotted between negative 360 and 540.

Influence of processing parameters on milling force of Z-Axis: a Influence of feed per tooth; b Influence of tool helix angle

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3 Finite Element Analysis of Milling of Inconel 718

In this study, a finite element model has been established, encompassing both the workpiece and the tool. Subsequently, the temperature and stress in the milling process are calculated.

3.1 Temperature Field Analysis of Milling of Inconel 718

During the milling process, the temperature at the tool tip, rake face, and flank face of the milling tool assumes paramount importance in the determination of tool wear and lifespan. Generally, it can be asserted that the higher the cutting temperature of the tool, the faster the tool will experience wear and eventual failure. Consequently, by means of simulating and analyzing the temperature profiles at the tool tip, rake face, and flank face, the tool’s performance and lifespan can be effectively predicted and assessed. To investigate the temperature variations at the tool tip, rake face, and flank face throughout the milling process, as depicted in Fig. 4, the finite elements situated within the tool tip (L1), rake face (L2), and flank face (L3) have been chosen, and a curve illustrating the time-dependent variations in the average temperature of these selected elements in L1, L2, and L3 has been generated.

Fig. 4
A graph of temperature versus time. 3 curves L1, L2, and L3 begin at (0, 20) and follow an increasing trend with some fluctuations between 20 and 700. L1 has the highest value, followed by L2 and L3. Several vertical lines are drawn between the curves. 3 F E M of L1, L2, and L3 are also given.

Temperature curves of each part of the tool

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3.2 Stress Field Analysis of Milling of Inconel 718

Difficult-to-machine materials, such as nickel-based superalloys, give rise to significant tool wear and thermal damage. Tool stress analysis enables the determination of both the maximum stress value and the distribution of stress within the tool, mitigating the risk of tool breakage or harm to the workpiece. Through an examination of the stress distribution within the tool, the stress-bearing regions and stress patterns can be ascertained. Furthermore, the analysis of tool stress unveils physical phenomena occurring during milling, including friction and thermal deformation, contributing to a more profound understanding of the machining process. To investigate the stress variations at the tool tip, rake face, and flank face throughout the milling process, as depicted in Fig. 5, the finite elements situated within the tool tip (L1), rake face (L2), and flank face (L3) have been chosen, and a curve illustrating the time-dependent variations in the average stress of these selected elements in L1, L2, and L3 has been generated.

Fig. 5
A graph of vonmises stress versus time. 3 curves begin from 0 to 2500, increase gradually, and follow a skewed distribution between 0 and 4000. L1 has the highest value, followed by L2 and L 3. 3 F E M of L1, L2, and L3 are also given.

Stress curves of each part of the tool

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3.3 Results of Analysis of Temperature Field and Stress Field

From the curve presented in Fig. 4, it is evident that during the progression of the milling process, the temperatures of the tool tip, rake face, and flank face exhibit an initial increase followed by a subsequent decrease. This phenomenon is associated with plastic deformation, accompanied by the generation of a substantial amount of cutting heat. The chips come into direct contact with the tool, transferring cutting heat into the tool, consequently resulting in a significant rise in tool temperature. As the chips eventually fracture and disengage from contact with the tool, a substantial portion of the cutting heat is carried away by the chips, leading to a reduction in tool temperature. In addition, it is observed that the temperature of the tool tip directly involved in the cutting process exceeds that of the rake face. This phenomenon arises because the tool tip is actively engaged in material removal, leading to significant plastic deformation and the generation of the highest amount of heat, resulting in the highest temperature, reaching a maximum of 682 °C. Furthermore, the temperature of the cutting edge’s rake face involved in the cutting process is higher than that of the flank face. This discrepancy is attributed to the pronounced extrusion and friction occurring between the rake face and the chips, which themselves possess elevated temperatures, leading to a sharp temperature increase. In contrast, the extrusion and friction experienced by the flank face against the machined surface are less severe, resulting in a lower temperature than that of the rake face. The maximum temperature recorded for the rake face is 614 °C, while the maximum temperature for the flank surface is 460 °C.

As shown in Fig. 5, the maximum stress region on the flank is larger than that on the rake face, with the stress in the section of the tool tip exceeding that on the flank face. Because the flank face of the tool is in contact with the workpiece’s surface, which means it experiences a significant amount of friction and resistance as the tool moves across the material. This results in higher stresses on the flank face compared to the rake face. The tool tip is where the actual material removal happens, and it experiences intense localized stress as it cuts through the superalloy. This stress can exceed that experienced by the flank face, especially if the cutting conditions are not optimized or if the tool is not designed for efficient chip formation. Additionally, the average stress on the flank is slightly higher than that on the rake face, which could be due to the cumulative effect of cutting forces, friction, and heat generated during machining. The rake face, while it does experience some stress, is primarily responsible for providing clearance and reducing the cutting forces.

4 Conclusions

In this study, an in-depth exploration of ball-end milling processes was undertaken, offering insights into the modelling of milling forces, finite element analysis of machining of Inconel 718, and the subsequent evaluation of temperature and stress within the tool. Significant findings underscored the pivotal role played by feed per tooth and tool helix angle in the determination of milling forces, with their impact on efficiency and tool performance being highlighted. The application of finite element analysis unveiled dynamic temperature and stress patterns during milling. Tool tip temperatures and stress exhibited peaks attributed to tool tip’s active engagement in material removal. This newfound comprehension aids in the mitigation of tool wear and damage.

In summary, this research advances our understanding of ball-end milling processes, allowing for more efficient machining and enhanced tool longevity, particularly in the context of challenging materials such as Inconel 718.