Force model for complex profile tool in broaching Inconel 718

Complex profile broaches are widely used in the manufacture of complex parts of aero-engines, but the forces in the broaching process are difficult to predict and control. A new numerical model for broaching force with complex profile tools was presented, which considered the area and arc length of the curved shear zone boundary. The area and arc length were calculated by the curve function of the boundary, which is firstly predicted by FEM simulation. Then, an experimental device was set up to carry out the broaching experiment with straight profile tools and complex profile tools in accordance with the progressive depth of the cut. Based on the experiments, the traditional broaching model and the modified model with the complex profile tool have been established. Compared with the traditional force model, the accuracy of the modified model has been moderately improved. Furthermore, the modified main broaching force (Y direction) model and the normal force (Z direction) model show a significant improvement in accuracy of 4.8% and 9.7%, respectively, which suggests that the projection area of curved shear zone A1 and the projection arc length of curved shear zone l1 have a big impact on the broaching process. It is firmly believed that the modified model proposed in this paper can provide guidance for the design of complex profile tools and facilitate the efficient and high-precision machining of complex parts.


Introduction
Inconel 718, as a kind of nickel-based superalloy, has been widely used in the manufacture of aero-engine and gas turbine components due to its high creep strength, comprehensive mechanical properties, and cavitation resistance [1][2][3][4]. However, the excellent performance of nickel-base superalloy has also brought huge challenges to its processing and manufacturing, especially during broaching of the turbine mortises. Broaching is still state of the art on account of its high toughness, high precision, and efficiency [5,6]. Nevertheless, in the processing of fir-tree profiles of the turbine disk, the excessive broaching force will bring about large deformation of the workpiece material, which will lead to the inaccuracy of the machining profile [7]. Controlling the broaching force to an appropriate value is an effective way to reduce tool wear and improve broaching quality. Hence, there is an urgent need to establish a more accurate cutting force prediction model in the machining process of nickelbase alloy with complex profile tools. [8]. Additionally, the establishment of a broaching force model in the process of complex tools is of great significance to assist companies in tool design and process parameter selection [9].
As for cutting modeling of complex profile tools, a great amount of work have been done by predecessors, and most of the models can be categorized as a theoretical analytical model [10,11], mechanistic model [12,13], empirical model [14,15], or finite element model [16,17]. But most scholars focus on the modeling of drilling or milling tools with complex blade profiles, while few scholars pay attention to broaching. Moreover, among the very limited literature on broaching force modeling, almost all focus on the modeling of broaching tools with a straight edge (the tooth profile is straight), For example, Kamath et al. [18] calibrated the broaching force coefficient of a straight edge broach with different materials such as medium carbon steel, aluminum, and cast iron in the broaching process. Sutherland et al. [7] modeled and verified the broaching process of keyway by oblique cutting model. Zhang et al. [19] simulated the broaching process of Inconel 718 alloy by 2D finite element method, the broaching process was numerically simulated, and the empirical model of broaching force with a straight profile was established. While there is little research on cutting modeling with complex profile tools. Cla et al. [20] considered the broaching with complex profile teeth, based on the oblique cutting theory, considering geometric parameters of the complex edge profile, the broaching model was established by discretizing the blade curve into several small segments and then summing it up. Fabre et al. [21] established a circular internal broaching force model with a complex profile through an empirical formula, in which the radial cutting force was predicted precisely. As can be seen, although a minority of scholars have paid attention to the broaching force modeling with complex profile tools, they still employ traditional models or methods to evaluate the broaching force with complex profile tools, but the special mechanism of chip formation is not paid attention to. It is believed that to further improve the accuracy of the cutting force prediction model, it is necessary to consider the complicated deformation of the shear zone caused by the complex profile tool.
For model modification, heretofore, with the vigorous development of computers, force modeling combined with finite element simulation has entered maturity. Based on the 2D finite element method, the friction data under heavy load and high temperature were introduced by Boyd et al. [22] to modify the finite element model and thus improve the modeling accuracy. Based on the coupled Eulerian-Lagrangian (CEL) finite element model, Peng et al. [23] considered the effect of the tool flank wear on the cutting force modeling, and the flank wear of the actual tool was measured by laser scanning microscope, then the geometric model was updated in the pretreatment of the simulation, so as to improve the cutting force modeling accuracy.
Based on the literature, in this paper, a novel broaching force model based on the projection area of the curved shear zone was proposed, and the 3D finite element simulation was employed to predict the area and arc length of the curved shear zone boundary. Based on the experimental results, the undetermined coefficients of the traditional and modified broaching force model are fitted. And the difference between the traditional model and the proposed one was also compared.

Analysis of complex profile tool
The complex profile broach refers to the broach for processing complex profiles such as the mortise of aero-engine turbine disks. It belongs to precision and complex cutting tools. Similar to traditional broaches, complex profile broaches are also divided into rough broaches, semi-precision broaches, and fine broaches. As shown in Fig. 1a, during broaching the mortises of turbine disk, the broaching process of complex profile tool can be equivalent to the orthogonal cutting of the curved blade tool. When the tool is fed the depth of δ, the curved cutting edge starts to contact the workpiece, but the whole edge of the tool does not contact the workpiece simultaneously. More importantly, the contact length of the edge profile is also variable during broaching, which results in a unique shear zone morphology. At the same time, as shown in Fig. 1c, the projection area of curved shear zone A 1 is on the XY plane. The area of the projection plane is related to broaching force, where the arc length of curved zone l 1 is also related to the broaching force. Hence, the area and arc length of the above two projection surfaces are used to characterize the broaching force. The complex profile tool is different from the traditional straight tool. In the first cut, for a straight cutting edge, the cutting depth (undeformed chip thickness δ s ) on the cutting edge is equal everywhere, but not for a complex profile cutting tool. For the complex profile tool, the cutting depth of the edge tip (purple pentagram) is taken as the undeformed chip thickness δ c , namely, the maximum cutting depth of a certain point of the edge. For the second or subsequent cutting with progressive cutting depth, whether it is a straight edge or a complex profile tool, the previous cutting trace line is equidistant from the current cutting edge, the cutting depth (undeformed chip thickness) at any point on the edge is equal. That is the distance along the Z-axis of the connecting line of the two tooltip points (blue and purple pentagram), as shown in Fig. 2.

Broaching force modeling process
According to Altintas et al. [24], the cutting process is the coexistence of the cutting effect and blade scraping effect. Thus, the broaching force model with a straight profile tool can be expressed by [20,25] where F t and F f are the main broaching force (Y-direction) and the normal force (Z direction), respectively, b and δ are the broaching width and broaching depth, namely, the rise per tooth of the broach, as shown in Fig. 3a; k ts and k fs are the specific cutting pressure of the main broaching force and the normal force, respectively, k tc and k fc are the edge force coefficients of the main broaching force and the normal force, α and β are constant.
As shown in Fig. 3b, being different from the straight profile tool, the undeformed area of the complex profile tool cannot be expressed in the form of A s = b * δ, but the area enclosed by the edge profile, denoted as A c . And the cutting width of the complex profile tool is not constant and will change with the progressive broaching depth.
Stage I: modeling of the undeformed cutting area According to the above analysis, the undeformed cutting area will vary with different broaching depths. As shown in Fig. 3c, d during the first broaching process, the undeformed area A c is a half-period sinusoid, but for the subsequent process, due to the previous cutting trace T i−1 , the current undeformed cutting area is surrounded by two sinusoids, shaped like a crescent, where the area is △A. Additionally, one of the sinusoids offsets equidistantly by the other, and the contact arc length of the edge profile l is the arc segment where the tooth contacts the workpiece. In this paper, the profile of the tool can be expressed by where A is the amplitude of sinusoidal function, ω is the circular frequency. The undeformed cutting area can be obtained through definite integral, which can be expressed as  Stage II: modeling of the traditional broaching force Similarly, the contact length of complex profile tool can be expressed by Consequently, the cutting area increment and contact length of complex profile tool change with the number of cuts that can be obtained.
According to the above analysis, the traditional broaching force model with complex profile tool can be expressed in the following form: where F t and F f are the main broaching force (Y direction) and the normal force (Z direction), respectively, A c is the undeformed area of the complex profile tool, l is the contact arc length of the complex profile tool.
Stage III: modeling of the modified broaching force From Fig. 3a, b, the projection area of curved shear zone A 1 and the arc length of curved shear zone l 1 are both distinctive. Due to the characteristics of this unique shear zone, the chip will be seriously deformed. Different from the traditional chip with uniform thickness, the chip generated through this special shear zone is thick in the middle and thin on both sides, and the deformation resistance of the chip will cause the change of broaching force. For this reason, the modified broaching force model with the complex profile tool should be given by where kʹ ts and kʹ fs are the shear zone area influenced coefficient of the main broaching force and the normal force, respectively, kʹ tc and kʹ fc are the shear zone arc profile influenced coefficient of the main broaching force and the normal force, and the acquisition of A 1 and l 1 will be presented in the next section.

Preprocessing of simulation
Based on the above analysis, it can be found that the area of curved shear zone A 1 and arc length of curved shear zone l 1 will be variable with feeding. To explore the chip formation and unique shear zone morphology, the broaching process simulation was realized through the 3D finite element method.
In the process of simulation, to control the element sizes and grid nodes in the cutting area to shorten simulation time, a relatively large depth of cut (0.4-0.6 mm) was taken, which almost would not affect the simulation accuracy. The workpiece was defined as an elastic-plastic body and was endowed with the JC constitutive model of Inconel 718, which was derived from literature [26], as listed in Table 1. Correspondingly, the tool was regarded as a rigid body, where the tool profile is defined as The number of grid nodes of the workpiece was 24,366, with 126,108 grid elements and the local mesh size of the workpiece is 0.005 mm. While relatively sparse grids could be used in the tool, where the number of grids nodes is 5046, with 26,088 grid elements, the local mesh size near the tooltip is 0.2 mm. Then the interaction between the tool and the workpiece was set, mainly the contact pair between the rake face and the bottom of the chip, and simultaneously the contact pair between the tool flank face and the machined surface, where the friction coefficient between the tool-chip contact surfaces was set to be 0.4. Then the bottom of the workpiece was fixed, and the moving speed of the tool was set to 2.4 m/min, as listed in Table 2. Additionally, the total simulation time is set to 0.1 s.
where A is the yield strength under quasi-static conditions; B is strain hard parameters; C is the strain rate hardening parameter; m is the thermal softening parameter; n is work hardening parameters; έ 0 is the quasi-static strain rate; T m is the melting point of the material; T r is room temperature.

Post processing of simulation
When the simulation was completed, the post-processing results of 0.6 mm depth were extracted first, the simulation state where the chip was just about to be raised from the surface (chip bulge) of the workpiece was captured. In the simulation environment, after a large number of observations, we found that most of the time, the boundary between the outside of the first light yellow crescent region and the inside of the second green crescent region is just the boundary of the chip bulge area (shear zone area) under the stress nephogram, as shown in Fig. 4a. Subsequently, the boundary of the curved shear zone of Fig. 4a was drawn point by point manually through CAD software, and 50 points were drawn at equal intervals starting from the origin O of each picture, to depict the contours of the shear zone, the position of the points of Fig. 4a in the CAD space was then derived, and the dimensional features of Fig. 4a was calibrated, respectively, where Fig. 4a was 1912:6 (that is, the actual size 6 mm in the simulation environment corresponds to 1912 mm in the CAD space). Then the scatter points plots of the profile of the curved shear zone was drawn and a fit curve based on the sine curve was obtained, as shown in Fig. 5, the blue line is the fitting curve obtained from the simulation at 0.6 mm depth of cut, which can be expressed as where y f is the boundary curve of the shear zone when the depth of cut is 0.6 mm. The fitting curve based on the sinusoidal curve can depict the morphology of the shear zone well. Hereafter, the broaching simulation at 0.5 and 0.4 mm depth of cut was carried out, and the similarity of the profile of the shear zone under each broaching depth was perceived, as shown in Fig. 4b, c, based on the fact that the boundary of curved shear zone at 0.4-0.6 mm depth of cut extended outward with an analogous shape, seemingly like water ripples spreading outward. A hypothesis that the profile of curved shear zone offset equidistantly with the progressive broaching depth was proposed. Similar to the method, the fitting plots of the profile of the curved shear zone at 0.4-0.5 mm depth of cut were obtained, as shown in Fig. 5. From top to bottom are the fitting curves of the simulated morphology of the curved shear zone where the depth of cut is 0.6 mm, 0.5 mm, and 0.4 mm, respectively. By least squares method, the fitting curve of the simulated morphology of curved shear zone at 0.5 mm depth of cut can be expressed as where y s is the boundary curve of the shear zone when the depth of cut is 0.5 mm while the fitting curve at 0.4 mm depth of cut can be expressed as where y t is the boundary curve of the shear zone when the depth of cut is 0.4 mm. Based on the above analysis, it can also be found that the maximum distance (Y-direction) on the boundary of the shear zone is about twice the depth of (9) y f = 1.3322sin 2 11.4706 ⋅ x + 6.3385  (4) to obtain the area and arc length.

Preliminary modeling work
According to Sect. 2.2, based on definite integral and curve integral. The undeformed area A c and contact arc length of the edge profile l were obtained with the progressive depth of cut as shown in Fig. 6. Based on the hypotheses aforementioned, the corresponding profile morphology of the curved shear zone at each broaching depth will be obtained based on the curve at broaching depth of 0.6 mm, as shown in Eq. (9).
The known interval depth of simulation is 0.1 mm, so the average offset of the profile can be calculated as where D 1 refers to the distance between the boundaries of the two shearing zones when the depth of cut differs by 0.1 mm. It can be inferred that the offset D 2 of the profile of the curved shear zone at the interval broaching 0.02 mm depth is The interval is exactly about twice the depth of the cut. Thus, based on the curve at 0.6 mm depth of cut, the broaching depth was successively shifted down by 0.02 mm. Similar to the undeformed cutting area A and the contact length of complex profile l, the projection area of the shear zone A 1 and the projection arc length of curved shear zone l 1 were (12)   Fig. 7. For the projection area of the shear zone A 1 , the situation at the first depth of cut is slightly different from the subsequent, as shown in the schematic in the lower right corner of Fig. 7, the projection area of the shear zone A 1 is the area surrounded by the profile of curved shear zone and the edge line of the workpiece. While for subsequent broaching, owing to the previous broaching trace, the current projection area △A 1i is the annular area surrounded by two profiles generated before and now, as shown in the schematic in the top left corner of Fig. 7.
At this point, each key variable in the mechanistic model has been obtained. With the corresponding experimental data, the parameters of the model can be determined, thus the model can be established.

Experimental device
To verify the modified broaching force model, a novel experimental system was set up, which mainly consisted of a driving slide platform, a three-dimensional dynamometer, and a cutter clamp device as shown in Fig. 8. The workpiece was fixed by a clamp, which was fixed on the three-phase force sensor by screws, and the sensor was fixed on the sliding platform and driven by a servo motor through a ball screw pair. Hereafter, the sliding platform carried the sensor and the clamping body, moving at the same time with the workpiece, and the tool was fixed in place thus generating relative speed to realize broaching. The maximum output force of 4000 N was provided by the servo motor through the ball screw pair.

Tool and workpiece material
The size of the workpiece was 30*20*14 mm, and the material of the workpiece was nickel-based superalloy Inconel 718. The detailed composition and mechanical properties of the material are listed in Tables 3 and 4. In this experiment, a customized tool was used, where the length of the blade of the tool b c is 13 mm, the edge width h 0 is 3 mm, with a 15° rake angle γ, and 6° relief angle α. The edge profile is a sinusoidal curve. It can be expressed as Eq. (7), as shown in Fig. 8. The tool was made of cemented carbide, whose material composition is shown in Table 5. The hardness of the tool material reaches up to 66HRC, with high wear resistance, good toughness, and high compressive strength. As a large amount of W, CR, and V elements are dissolved in the tempered martensite after quenching, the tempered martensite maintains a high hardness, and the precipitated carbides accumulate slowly, so the tempered martensite can still maintain strength at high temperature, which enables it to be widely used in broaching of difficult-to-cut materials.

Experimental procedure
To calibrate coefficients in Eq. (1), the straight profile broaching experiment is carried out. The rake angle of the straight profile tool γ is 15°, and the relief angle α is 6°, and the width b s is 13 mm. As shown in Fig. 9, to change the   Table 6, where tests were repeated 3 times. Then, a single-factor broaching experiment with a complex profile tool was carried out with a fixed broaching speed of 2.4 m/min, and the progressive depth of cut was 0.02 mm each time, 15 times of progressive broaching are completed, with a total of 0.3 mm depth of cut. The details of broaching parameters can be seen in Table 7, where tests were repeated 3 times for validating dates. The progressive depth of cut was realized by placing a gasket at the bottom of the workpiece step by step, thus raising the workpiece by 0.02 mm each time the above process parameters are all the recommended in actual turbine disk tenon groove broaching. The broaching force in two directions was obtained by a ME (Type ME-K3D120) sensor with a charge-amplifier (Type GSV-1A4) and a data acquisition system for force measurement (Type INV3018CT) connected to the computer. The experimental data were collected in real-time through UTEK data collection software, under a sampling frequency of 10 kHz.

Establishment of the traditional model and modified model
The coefficients of the main broaching force of the straight profile tool were obtained by least-squares fitting, where k ts , α, and k tc are 6441, 1.20, and 22.02, respectively. The coefficient of determination (R) of this fitting is 0.9976, with a moderate root-mean-square error (RMSE), as shown in TS-X in Table 8, indicating a high degree of the fitting. Accordingly, these coefficients were adopted in the main broaching force model with a complex profile tool and corresponding modified model. Then based on coefficients obtained by broaching with straight profile tool, combining with the change law of the undeformed cutting area A c and the contact length of complex profile l with cutting times (namely the broaching depth) obtained in Sect. 3.3, the relationship between main broaching force with the undeformed cutting area A c and the contact length of complex profile l can be established, namely the main broaching force model with  . 9 Schematic diagram of broaching with a straight profile tool  complex profile tool, which in essence, is the modification of the main broaching force model based on the traditional model. And the information of the traditional model of the main broaching force (Y-direction) with complex profile tool is shown in TC-X in Table 8. It can be found that directly applying the traditional model to the main broaching force model with a complex profile tool does not work well. By introducing features of the curved shear zone, the modified main broaching force model was established, where k' tc and k' tc are 606.5 and 7.89, respectively, as shown in MC-X in Table 8. Among the 15 groups of experimental data, the first 10 groups were used for fitting (broaching depth from 0.02 to 0.20 mm), with the last 5 groups used for verification (broaching depth from 0.2 to 0.30 mm). Intuitively, it can be seen the projection area of the shear zone A 1 plays an important role in the improvement of model accuracy while the projection arc length of the shear zone l 1 almost has no effect on the model.
For the normal force, the modeling process is similar to the main broaching force. Firstly, the model coefficients k fs , β, and k fc were calibrated by broaching with a straight profile tool, as shown in TS-Z in Table 8. After that, according to the above coefficients, combined with the undeformed cutting area A c and the contact length of complex profile l, the traditional normal force model with a complex profile tool was established, as shown in TC-Z in Table 8. Ultimately, by introducing the character of the curved shear zone, the modified main broaching force model was established, where k' fs and k' fc are 42.47 and 20.74, respectively, as shown in MC-Z in Table 8. Among the 15 groups of experimental data, the first 10 groups were used for fitting (broaching depth from 0.02 to 0.20 mm), and the last 5 groups were used for verification (broaching depth from 0.22 to 0.30 mm). It can be seen that both the undeformed cutting area A c and the contact length of edge profile l have effects on the model accuracy, thus, lower down the root-mean-square error (RMSE) of the normal force model.

Comparison of traditional model and modified model
It can be seen from the mean square error, for the main broaching force, the modified model based on the characteristics of the curved shear zone shows a huge reduction in root-mean-square error (RMSE) from 127.1 to 49.58. And the goodness of fit also shows a slight increase, which is reflected in the increase in R-square. Moreover, a set of validation data were used to test the generalization ability of the fitting. Concretely, the last five sets of experimental results were used for verification as shown in Table 9, the maximum error of the traditional model is 264.4 N, the minimum is 115.3 N, and the average is 178.4 N, while the maximum error of the modified model is 148.7 N, the minimum is 2.8 N, and the average is 62.6 N, the average predicted value is 115.8 N lower than that of the traditional model. And the relative error rate is selected to evaluate the fitting effect. As shown in Fig. 10, the upper green line is the relative error of fitting of the traditional main broaching model with a complex profile tool, while the yellow one is the modified model. The relative error rate of the traditional main broaching force model was 10.96% at the maximum and 4.78% at the minimum, correspondingly, for the modified model, the maximum relative error rate is 6.16%, with the minimum relative error rate of 0.12%.
According to the relative error information above, it can be concluded that compared with the traditional main broaching force model with a complex profile tool, the mean error rate of the modified main broaching force model with a complex profile reduces by 4.8%. This suggests that the accuracy of the modified model is moderately improved compared to the traditional model, thus, indicating that the features of curved shear zone such as the projection area of curved shear zone A 1 and the projection arc length of curved shear zone l 1 have a huge impact on the broaching with a complex profile tool. Similarly, compared with the traditional model with a complex profile tool, the RMSE of the modified normal force based on the characteristics of the curved shear zone is greatly reduced, about a third of the traditional one. Furthermore, a set of validation data which were from the same source as the main broaching force were used to test the generalization ability of the fitting as shown in Table 10  the average predicted value is 80 N lower than that of the traditional model. In like manner, the relative error rate is selected to evaluate the fitting effect. As shown in Fig. 11, the upper cyan line is the relative error of fitting of traditional normal force model with complex profile tool while the red one is modified, model. The relative error rate of the traditional normal force model was 16.78% at the maximum and 9.06% at the minimum, correspondingly, for the modified model, the maximum relative error rate is 6.00%, with the minimum relative error rate of 0.30%. By analysis, compared with the traditional normal force model with complex profile tool, the mean error rate of the modified normal force model with complex profile tool reduces 9.7%. It can be concluded that the accuracy of the modified model is greatly improved compared to the traditional model, thus certificating again that the features of curved shear zone such as the projection area of curved shear zone A 1 and the projection arc length of curved shear zone l 1 have a huge impact on the broaching with complex profile tool.

Conclusion
In this paper, a novel broaching force model with the projection of shear zone is proposed, which introduces the morphological characteristic of the curved shear zone to modify the traditional main broaching force (Y direction) and the normal force (Z direction) models, so as to predict the progressive broaching process of Inconel 718 with complex profile tool. At length, the effectiveness of the model is verified by comparing it with the traditional broaching model. The main findings of the paper can be summarized as follows: • The shear deformation zone when broaching with complex profile tool was explored, which is a great difference from the traditional straight profile tool. The morphology and the shear zone of the complex profile tool are studied through broaching simulation. • Experimental results show that the projection area of the curved shear zone A 1 and the arc length of the curved shear zone boundary L 1 were of great importance in broaching force modeling.
Compared with the traditional broaching model, the modified model based on the curved shear zone was first introduced to predict the progressive broaching process of Inconel 718 with a complex profile tool. The results show that in the direction of main broaching force (Y direction), compared with the traditional main broaching force model with complex profile tool, the relative error rate of the modified broaching force model with complex profile tool is reduced by 4.8%, and the average predictive value is 115.8 N lower than that of the traditional model. Its relative error rate in the normal force (Z direction) is reduced by 9.7%, and the average predicted value is 80 N lower than that of the traditional model. which shows the validity of the modified model. Moreover, the modified model proposed in this paper can provide guidance for the design of complex profile tools and facilitate the efficient and high precision machining of complex parts.