## 1. Introduction

Titanium alloys due to their unique mechanical properties are widely used in industrial applications. Especially, the Ti6Al4V alloy is one of the most widely used alloys and constitutes more than 50% of titanium products globally [

1]. Most applications of this alloy are found in the aerospace industry, but it is also used for the production of medical devices due to its biocompatibility. However, it exhibits particular attributes as a material, such as low thermal conductivity, high strength at elevated temperatures, and low modulus of elasticity which lead to low machinability ratings [

2,

3]. Machining is one of the most prevalent processes for the production of industrial components and, due to the reasons mentioned above, cutting of titanium is usually accompanied by high cutting forces and high cutting temperatures. These major factors lead to excessive tool wear, shortened tool life, and poor surface quality of the final workpiece.

In practice, to increase productivity and at the same time keep quality at a high level, it is always necessary for the proper machining parameters to be chosen. There are mainly two ways that this goal can be attained. The first one involves conducting experiments to accumulate knowledge on the dependence of machining conditions upon important physical quantities (cutting forces, temperature etc.) during cutting. The second way involves the simulation of the machining process through the development of computational models. During the last decades, simulations have become increasingly popular and allow researchers to analyze, study, and understand in depth the physics of machining.

The finite element method (FEM) is one of the most frequently used numerical techniques for engineering simulations. However, there are a number of difficulties that arise when it is used for cutting simulations. Firstly, large strains lead to severe element distortions and consequently to the termination of the simulation, due to negative volume elements. Secondly, in order to model the fracture of the workpiece, a failure criterion has to be implemented into the code, so that certain elements are deleted from the grid when certain criteria are satisfied. The problem of distorted elements is usually solved with remeshing algorithms, which recreate the mesh after a user-defined number of time steps, resulting in increased computational cost. An alternative solution to this problem is based on element erosion/deletion when the stresses or strains inside an element surpass a certain threshold. The disadvantage of this approach is the fact that many elements can be artificially deleted from the grid, leading to non-physical simulations.

In [

4], the orthogonal cutting is simulated by the FEM where a simple geometric criterion, combined with a mesh rezoning algorithm is used for the modeling of material separation. In [

5,

6,

7], fracture is simulated by an element deletion algorithm which is based on a fracture criterion. Also, avoidance of severe element distortion is overcome by the usage of remeshing algorithms. In [

8,

9,

10], material separation in the simulations is modeled by the formation of adiabatic shear bands and the usage of fracture criterions is avoided. However, the usage of remeshing algorithms still remains a necessity for the same reasons. In [

11], the influence of cutting conditions on the stress at the rear surface of the tool is investigated. In [

12], a mathematical model for the calculation of cutting forces is developed, based on the results of a FEM model. In [

13,

14], the influence of the constitutive model and the damage criteria on the prediction of cutting forces is studied. An alternative to classical FEM techniques is the use of combined Lagrangian–Eulerian formulations, where the grid is not tied on the workpiece but it is fixed in space [

15,

16,

17].

During recent years, meshless numerical methods are gaining increased popularity because the particles/nodes are not strictly connected with their neighbor particles, but are relatively free to move in space, unlike the nodes of a finite element grid. In this study, the focus is on the smooth particle hydrodynamics meshless method. The SPH method was developed in the 1980s and was used for the numerical simulation of physical problems that belong to the field of astrophysics [

18] and a decade later was modulated for solid mechanics problems [

19]. A comprehensive analysis of the SPH method and examples regarding its applications in solid and fluid mechanics can be found in [

20,

21].

One of the earliest implementations of the SPH method to orthogonal cutting simulations can be found in [

22], where the benefits of the method are presented in comparison to the traditional FEM approach. In [

23], a 2D model is presented where friction is predicted in workpiece/cutting tool interface without the usage of a friction model, since both bodies are modeled with SPH particles. In [

24], a model of the same type is used for the prediction of the variation of cutting forces caused by a worn cutting tool. In [

25,

26,

27,

28,

29,

30], SPH models of orthogonal, or oblique, cutting are created where the cutting tool is modeled with finite elements and the workpiece with SPH particles, which means that a friction model had to be used. In [

30], the effect of subsequent cuts on the evolution of residual stresses in the workpiece is presented. Studies of SPH simulations regarding the influence of factors describing the material behavior such as the friction coefficient, the equation of state, or the usage of a damage-evolution criterion can be found in [

31,

32,

33]. The influence of more intrinsic parameters to the SPH method, such as timestep and particle density, can be found in [

34]. Similarly, in [

35] the effect of important SPH control parameters is examined.

One of the deficiencies of the SPH method in metal cutting simulations is the inability of prediction of a realistic chip curvature, as it is mentioned in [

36]. For this reason, alternative formulations are proposed to the standard SPH scheme, where this issue is resolved. One of these formulations is called renormalization and more details about its foundations and development can be found in [

37,

38]. Orthogonal cutting simulations with SPH can be found in the literature [

23,

24,

25,

26,

27,

28,

29,

30,

31,

32,

33,

34,

35] for different formulations. However, there are no studies that examine the influence of the chosen formulation, on the prediction of the cutting forces. The aim of this study is the investigation of the differences in the prediction of cutting forces between two different formulations. For this purpose, numerical simulations of orthogonal cutting of Ti6Al4V titanium alloy are presented, where both formulations are used, namely the standard and the renormalized. This parametric analysis is complemented by conducting numerical simulations with different particle densities.

In

Section 2, the mathematical foundations of SPH are summarized. The presentation of the theory in this section is based on [

21]. In

Section 3, the main formulas describing the material behavior are also provided, while in

Section 4, the configuration of the numerical simulation is described. Furthermore, the way the parametric analysis was conducted, is described in detail. In

Section 5, the main results of the parametric analysis are presented. For all the simulations, the commercial solver LSDYNA [

39] was used.

## 4. Numerical Simulation Configuration

Due to its relative simplicity, most published research is concerned with the simulation of orthogonal cutting. However, the results of a simulation need to be validated by actual experimental data and orthogonal cutting is not a widely used process in practice. Therefore, in many studies experimental data is derived from actual machining operations. Machining processes are mainly 3-dimensional and can be approximated by a 2-dimensional process to a certain extent. In this study, experimental results from reference [

43] are used, where actual orthogonal cutting is conducted.

In the experimental reference, the cutting tool is moving with constant speed V into a flat strip of width W

_{c} = 1 mm, height H = 2 mm and a length L = 10 mm. The experiment is repeated for three depths of cut h

_{c} = 0.06, 0.04 and 0.1 mm. The material of the cutting tool is WC/Co tungsten carbide and of the workpiece Ti6Al4V titanium alloy (

Table 2). The experimental conditions are summarized in

Table 3.

The simulation set up can be seen in

Figure 2. The height of the workpiece H is three times the depth of cut (H = 3 h

_{c}) and the width of the strip W

_{c} is 0.01 mm. The length of the workpiece L is 1.5 mm, when the depth of cut is h

_{c} = 0.1 mm and is reduced to 1 mm for the remaining two values of h

_{c}. For the cutting tool and for the lower part of the workpiece up to one third its height, finite elements are used, and the rest of the workpiece is modeled with SPH particles. The nodes at the bottom and the edge of the strip are constrained in all directions.

The simulation of the experiment with the actual strip length is, from a computational standpoint, time-consuming and also unnecessary since the cutting process transitions to a steady state, where the cutting forces are stabilized long before the cutting tool reaches the end of the strip. The same applies for the width of the workpiece because all nodes are constrained to the z direction and are free to move in the x-y plane; it is redundant for the actual width of the workpiece to be modeled and scaling of the cutting forces to a factor of 100 is preferable. Finally, the SPH method is computationally more demanding than the standard FEM, and this is the reason why it is usually preferred for the area close to the path of the cutting tool to be modeled with particles. For the other parts, finite elements are considered more suitable and the mesh becomes gradually coarser at the bottom of the workpiece.

The parametric analysis is conducted based upon two parameters, the SPH formulation and the density of the particles. Initially, the distance d between the particles and the depth of cut h_{c} are fixed to a certain value and the simulation is carried out twice. Initially, a standard SPH formulation is used, followed by a renormalized formulation.

Consequently, the simulation with the renormalized formulation is carried out once again and the particle distance d is reduced to half of the initial value. Afterward, a different value for the depth of cut h_{c} is selected and the same procedure is repeated.

## 5. Results and Discussion

In

Figure 3, the equivalent stress contours are depicted for a fixed depth of cut h

_{c} = 0.04 mm. In

Figure 4 and

Figure 5, the stress contours are shown for depths of cut of h

_{c} = 0.06 and 0.1 mm, respectively. By comparing the frames a and b for each depth of cut, an obvious observation is the chip curvature. By using the standard SPH formulation, a straight numerical chip is produced that follows the chip face of the cutting tool. The usage of the renormalized formulation leads to the production of a more realistic chip. The nonrealistic chip curvature of the standard formulation is attributed to the absence of particles in the exterior areas of the boundaries of the solution domain (workpiece) [

36]. This absence leads to non-approximate calculations near the boundaries.

As can be seen in

Figure 3b,c, the chip curvature is increased for a greater particle density and the same observation applies and for the other two values of the depth of cut. In contrast, the maximum stress is not affected considerably, regardless of the depth of cut, the formulation or the density of the particles. This can be explained by the fact that the maximum stress that is induced on the workpiece in the primary shear stress zone is dependent on the strength of the material.

In

Figure 6, the plastic strain contours are depicted for each value of the depth of cut. In this figure, only the chips that are produced by the renormalized SPH are chosen, with the greater particle density. The areas with the larger values of strain can be seen at the secondary shear zone at the interface between the cutting tool and the chip, at the tertiary shear zone where the cutting tool rubs the surface of the workpiece, and also at the shear zones that are created at the primary shear zone.

In

Figure 7,

Figure 8 and

Figure 9, the cutting force F

_{c} and the feed force F

_{f} are depicted in relation to time. Each figure corresponds to one of the three values for the depth of cut. Also, as can be seen from these figures, the forces start to stabilize at the time of 0.1 to 0.3 ms. For the values depicted in each of these figures, an average value is computed from the time that the forces remain relatively stable until the end of the simulation. The average values are compared to the experimental ones.

As shown in

Figure 7a, the results of the standard SPH scheme are not satisfactory since the predicted values are less than half the values of the experimental ones. In contrast, the values predicted by the renormalized formulation, which are depicted in

Figure 7b, are very close to the experimental ones. The cutting force F

_{c} and feed force F

_{f} are overpredicted with an error of about 7.56% and 4.39%, respectively. By comparing

Figure 7b with

Figure 7c, it is obvious that the forces predicted by the model with greater particle density are more stable and oscillate less. In addition, lower values are predicted for the cutting force F

_{c} and there is also a small increase in the feed force F

_{f}. The cutting force F

_{c} is underpredicted with an error of about 15% and the feed force F

_{f} is overpredicted with an error of 6.8%.

As can be seen in

Figure 8a and

Figure 9a, the results produced from the standard SPH formulation remain far from the experimental values regardless of the depth of cut. By comparing

Figure 8b to

Figure 8c and

Figure 9b to

Figure 9c, the same pattern is observed. The models with greater particle density provide relatively decreased values for the cutting force F

_{c} and increased values for the feed force F

_{f}. Also, especially for the cutting forces, there is less variation between the maximum and minimum values. For a depth of cut of h

_{c} = 0.06 mm, the cutting force F

_{c} is overpredicted with an error of 7.9%, the feed force F

_{f} is underpredicted with a divergence of 20% from the experimental value. By increasing the number of particles, the predicted value of the cutting force F

_{c} is lower than the experimental and the error is increased slightly to 9%. The predicted value for the feed force F

_{f} is increased and almost coincides with the experimental value. For a depth of cut of h

_{c} = 0.01 mm, as can be seen in

Figure 9, the averaged predicted cutting force F

_{c} is very close to the experimental value and they differentiate by less than 1.2%. However, the model which is comprised of a greater number of particles underpredicts the cutting force F

_{c} with an error of 10.4%. For both models, as can be seen from

Figure 9b,c, the feed force F

_{f} is underpredicted and the averaged values differ from the experimental with a percentage of 25%.

The numerically predicted cutting and feed forces are gathered in

Table 4 and

Table 5. At this point, it is obvious that only models where the renormalized formulation is used provide results close to the experimental values. Also, there is a negative correlation between cutting forces F

_{c} and particle density. Models with smaller particle density estimate the cutting forces more accurately, which is an indication that the SPH models might converge to a value different from the experimental; still, the maximum deviation does not exceed 15%. For the feed forces F

_{f}, a clear pattern is not visible. Although there is a positive correlation between feed force and particle density, the sensitivity is almost zero for h

_{c} = 0.1 mm and there is a slight change of the predicted values for h

_{c} = 0.04 mm, but for h

_{c} = 0.06 mm, the deviation between the feed forces estimated by the different particle density models approaches 26%. Generally, the cutting force F

_{c} is estimated more accurately comparing to the feed force F

_{f} and this can be attributed to the parameters chosen for the J–K model. As mentioned in reference [

42], parameters provided in the literature for the J–K model usually do not lead to accurate estimations of both cutting and feed forces as well.