# Finite Element Investigation on Cutting Force and Residual Stress in 3D Elliptical Vibration Cutting Ti6Al4V

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## Abstract

**:**

## 1. Introduction

## 2. Kinematic of 3D-EVC

_{0}, from which the tool cuts the surface remaining from the previous cycle until point t

_{1}, which is the lowest point of the trajectory in the y direction, where the tool velocity is zero. The tool leaves the workpiece material at point t

_{2}, ending the effective cutting stage. The next cycle of cutting starts at point t

_{3}.

_{x}, φ

_{y}and φ

_{z}are the initial phase angles in the x, y and z directions. v

_{c}is the cutting speed. As shown in Figure 2, by synthesizing the trajectory equations of the three axes, the spatial trajectory of 3D-EVC can be obtained. In addition, from the projection of each direction in Figure 2, it can be seen that 3D-EVC is reciprocating on the three projection planes.

## 3. Finite Element Method

#### 3.1. Material Constitutive Model

_{m}is the melting temperature of the workpiece and T

_{r}is the ambient temperature. In this paper, the parameters of Johnson-Cook constitutive model are shown in Table 1.

#### 3.2. The Criterion of Material Failure

_{1}~d

_{5}are the material failure constants, as shown in Table 2.

_{0}is the element feature length in the simulation model, σ

_{y}

_{0}is the yield stress at initial damage, $\overline{u}$ is the equivalent plastic displacement of material failure process, ${\overline{u}}_{f}$ is the equivalent plastic displacement when the material fails completely.

_{0}of the simulation model. Therefore, the initial failure displacement value is determined according to L

_{0}. Then, according to the comparison between the cutting force in the simulation and the cutting force in the experiment, the appropriate failure displacement value is finally obtained.

#### 3.3. The Establishment of Finite Element Model

## 4. Results and Discussions

#### 4.1. Effect of Cutting Parameters on Cutting Forces

_{p}, F

_{t}and F

_{n}are the principal force, thrust force and normal force respectively [41].

#### 4.1.1. Effect of Cutting Speed on Cutting Force

#### 4.1.2. Effect of Vibration Amplitude on Cutting Force

_{c}/2πfa). With the increase of amplitude a, the effective cutting time decreases in a cutting cycle. It can be seen from Figure 6d that the average principal force is 69.32 N at 8 μm, and 34.75 N at 23 μm, decreasing by 49.9%.

#### 4.1.3. Effect of Vibration Frequency on Cutting Force

#### 4.1.4. Effect of Cutting Depth on Cutting Force

#### 4.2. Effect of Cutting Parameters on Residual Stress

#### 4.2.1. Effect of Cutting Speed on Residual Stress

#### 4.2.2. Effect of Vibration Amplitude on Residual Stress

#### 4.2.3. Effect of Vibration Frequency on Residual Stress

#### 4.2.4. Effect of Cutting Depth on Residual Stress

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The trajectory and its projection of tool in 3D-EVC (

**a**) 3D view (

**b**) the projection of trajectory in XoZ plane (

**c**) the projection of trajectory in XoY plane (

**d**) the projection of trajectory in YoZ plane.

**Figure 4.**Analysis of the cutting force of the cutting tool acting on the workpiece (

**a**) the stage similar to ordinary cutting (

**b**) the stage of friction reversal.

**Figure 5.**Results of cutting force and TOC with various cutting speeds (

**a**) principal force (

**b**) thrust force (

**c**) normal force (

**d**) average cutting force (

**e**) TOC with various cutting speeds in XoY plane (

**f**) TOC with various cutting speeds in XoZ plane.

**Figure 6.**Results of cutting force and TOC with various amplitudes a (

**a**) principal force (

**b**) thrust force (

**c**) normal force (

**d**) average cutting force (

**e**) TOC with various amplitudes a in XoY plane (

**f**) TOC with various amplitudes a in XoZ plane.

**Figure 7.**Results of cutting force and tool trajectories with various amplitudes b (

**a**) principal force (

**b**) thrust force (

**c**) normal force (

**d**) average cutting force (

**e**) cutting trajectory with various amplitudes b in XoY plane (

**f**) cutting trajectory with various amplitudes b in XoZ plane.

**Figure 8.**Results of cutting force and tool trajectories with various amplitudes c (

**a**) principal force (

**b**) thrust force (

**c**) normal force (

**d**) average cutting force (

**e**) cutting trajectory with various amplitudes c in XoY plane (

**f**) cutting trajectory with various amplitudes c in XoZ plane.

**Figure 9.**Results of cutting force and TOC with various vibration frequencies (

**a**) principal force (

**b**) thrust force (

**c**) normal force (

**d**) average cutting force (

**e**) TOC with various vibration frequencies in XoY plane (

**f**) TOC with various vibration frequencies in XoZ plane.

**Figure 10.**Results of cutting force with various cutting depths (

**a**) principal force (

**b**) thrust force (

**c**) normal force (

**d**) average cutting force.

**Figure 16.**Results of residual stress with various vibration frequencies (

**a**) stress-xx (

**b**) stress-zz.

**Table 1.**Johnson-Cook parameters for Ti6Al4V alloy [38].

A (MPa) | B (MPa) | C | n | m | T_{m} (°C) | T_{r} (°C) |
---|---|---|---|---|---|---|

782.7 | 498.4 | 0.028 | 0.28 | 1 | 1660 | 20 |

**Table 2.**Johnson-Cook damage parameters [39].

d_{1} | d_{2} | d_{3} | d_{4} | d_{5} | $\dot{{\mathit{\epsilon}}_{0}}\left({\mathbf{s}}^{-1}\right)$ |
---|---|---|---|---|---|

−0.09 | 0.25 | −0.5 | 0.014 | 3.87 | 0.001 |

**Table 3.**The material parameters of tool and workpiece [39].

Properties | Ti6Al4V | PCD |
---|---|---|

Density (kg/m^{3}) | 4440 | 14,450 |

Young’s modulus (GPa) | 119 | 640 |

Poisson’s ratio | 0.29 | 0.22 |

Specific heat (J/kg/K) | 661 | 220 |

Thermal conductivity (W/m/K) | 6.8 | 75.4 |

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**MDPI and ACS Style**

Li, S.; Han, J.; Yu, H.; Wang, J.; Lu, M.; Tian, Y.; Lin, J.
Finite Element Investigation on Cutting Force and Residual Stress in 3D Elliptical Vibration Cutting Ti6Al4V. *Micromachines* **2022**, *13*, 1278.
https://doi.org/10.3390/mi13081278

**AMA Style**

Li S, Han J, Yu H, Wang J, Lu M, Tian Y, Lin J.
Finite Element Investigation on Cutting Force and Residual Stress in 3D Elliptical Vibration Cutting Ti6Al4V. *Micromachines*. 2022; 13(8):1278.
https://doi.org/10.3390/mi13081278

**Chicago/Turabian Style**

Li, Shiyu, Jinguo Han, Haiqiang Yu, Jinhui Wang, Mingming Lu, Yebing Tian, and Jieqiong Lin.
2022. "Finite Element Investigation on Cutting Force and Residual Stress in 3D Elliptical Vibration Cutting Ti6Al4V" *Micromachines* 13, no. 8: 1278.
https://doi.org/10.3390/mi13081278