# 3D-FEM Approach of AISI-52100 Hard Turning: Modelling of Cutting Forces and Cutting Condition Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. CAD-Based Setup of the Machining Process

_{t}corresponds to the tangential force, F

_{r}is the radial force and F

_{a}represents the feed force. Additionally, Figure 1a illustrates the contact point of the tool on the cutting surface with the aid of two extra schematics, revealing the lead angle, the rake angle, as well as the inclination angle. The values of these angles are drawn from the geometry of both the tool-holder and the insert. According to the specifications of the tool-holder and the insert, the lead angle was 75°, whereas the rake and the inclination angles were both equal to −6°. Furthermore, Figure 1b illustrates the geometry and the characteristics of the turning inserts used. The face land width and the face angle are the same for all three inserts, and were equal to 0.10 mm and 20°, respectively. Moreover, the tool nose radius (r

_{ε}) was 0.80 mm, 1.20 mm and 1.60 mm, according to the ISO code of each insert. Lastly, the SNGA-ceramic inserts are typically used for machining cast iron and hardened steel.

_{c}), the feed (f), the depth of cut (a

_{p}) and the tool nose radius (r

_{ε}), each in three levels.

#### 2.2. Pre-Processing of the Numerical Model

#### 2.2.1. Analysis Interface Specifications

_{conv}= 0.02 N/(s × mm × °C) and h

_{cond}= 45 N/(s × mm × °C), respectively.

#### 2.2.2. Material Modelling

_{c}is the critical value of the damage during fracture; σ

_{max}and $\overline{\sigma}$ represent the maximum tensile principal stress and the effective stress, respectively; ε

_{f}denotes the limit fracture strain; ε

_{pl}is the plastic strain.

_{f}is the frictional shear stress; μ denotes the shear friction coefficient, whereas σ

_{n}represents the stress developed on the tool-chip interface. A number of similar studies on the machining of different types of steel [35,36] suggest applying a friction coefficient between −0.5 and 0.6. Moreover, the friction coefficient value that DEFORM™ proposes is 0.6, which was also the selected value for the simulation runs included in the present study.

## 3. Results and Discussion

#### 3.1. FEM-Based Evaluation of the Resultant Cutting Force

_{resultant}is the resultant machining force in N, F

_{t}represents the tangential force in N, F

_{r}is the radial force in N and F

_{a}denotes the feed force in N.

#### 3.2. Mathematical Modelling of the Resultant Cutting Force

_{c}), feed (f), depth of cut (a

_{p}) and tool nose radius (r

_{ε}), whereas the response (output) is the resultant cutting force (F

_{resultant}).

_{0}denotes the fixed term, X

_{i}represent the input variables (cutting speed, feed rate, depth of cut and tool nose radius) and, finally, b

_{i}, b

_{ij}, and b

_{ii}relate to the vectors that contain the regression coefficients (linear, quadratic, and cross-product, accordingly).

#### 3.3. Analysis and Validation of Mathematical Model

_{c}, f, a

_{p}, r

_{ε}, f

^{2}, a

_{p}

^{2}and f × a

_{p}contribute the most to the model, indicating the strong influence of the four cutting parameters on the generated forces, especially of the feed and the depth of cut. Additionally, the variation of the response, can be graphically presented with the aid of the graph of Figure 6a, as well as expressed with the total sum of squares. Finally, the p-value (zero) of the model itself indicates that the chances of producing abnormal results are rather low.

#### 3.4. Investigation of the Cutting Parameters’ Influence

_{resultant}. Figure 7b depicts the combined effect of cutting speed and feed, which is particularly strong as feed increases, especially after 0.11 mm/rev. An exception occurs for feed between 0.08 mm/rev and 0.11 mm/rev, where the generated force is slightly decreased. On the other hand, as cutting speed increases, the produced cutting force exhibits a small decrease. Figure 7c illustrates how cutting speed and depth of cut affect the produced machining force. Similar to the two previous cases, the influence of the cutting speed is not significant. In contrast, depth of cut has a great impact on F

_{resultant}. Quick growth is noted in the resultant cutting force between a

_{p}= 0.20 mm and a

_{p}= 0.25 mm, which decreases for each step of depth of cut. Depth of cut and feed affect the produced cutting force the most; their combined effect is illustrated in Figure 7d. The surface of Figure 7e illustrates how feed and tool nose radius influence the resultant machining force. Finally, Figure 7f depicts the impact of depth of cut and tool nose radius on the generated force.

#### 3.5. Optimization Process

#### 3.6. Confirmation of Mathematical Model

## 4. Conclusions

- Both the established FE model and the mathematical one can predict the generated cutting forces with acceptable errors. The relative error found for the comparison between the numerical and the experimental results ranged between −1.7% and 16.7%, whereas the numerical and the statistical results ranged between −7.9% and 11.3%.
- Especially with the use of the mathematical model, future experimental testing can be skipped and instant results can be delivered for F
_{resultant}within the range of the investigated parameters. - It was revealed that both depth of cut and feed increasingly act on the generated force, especially depth of cut. The increase percentage when shifting from level one value to level three for feed and depth of cut, is close to 35% and 78%, respectively. Tool nose radius also seems to have an increasing effect, but of no significance, at least compared to the other two parameters. On the other hand, cutting speed seems to lower the produced forces by a small, but not negligible, amount.
- Finally, the optimal cutting conditions were found for three different cutting inserts. Namely, 175.76 m/min cutting speed, 0.097 mm/rev feed and 0.20 mm depth of cut for the 0.80 mm tool, 177.78 m/min cutting speed, 0.098 mm/rev feed and 0.20 mm depth of cut for the 1.20 mm tool and, lastly, 199.73 m/min cutting speed, 0.082 mm/rev feed and 0.20 mm depth of cut for the 1.60 mm tool.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The CAD-based setup of the tool-workpiece assembly and (

**b**) the geometry of the turning insert.

**Figure 3.**The flow stress diagram [28] for AISI-52100 steel at 20 °C.

**Figure 6.**Residual analysis graphs: (

**a**) probability plot, (

**b**) residuals versus fitted values, (

**c**) residual histogram and (

**d**) residuals versus order.

**Figure 7.**3D surface plots: (

**a**) F

_{resultant}vs. V

_{c}, r

_{ε}, (

**b**) F

_{resultant}vs. V

_{c}, f, (

**c**) F

_{resultant}vs. V

_{c}, a

_{p}, (

**d**) F

_{resultant}vs. f, a

_{p}, (

**e**) F

_{resultant}vs. f, r

_{ε}and (

**f**) F

_{resultant}vs. a

_{p}, r

_{ε}.

Level | V_{c} [m/min] | f [mm/rev] | a_{p} [mm] | r_{ε} [mm] |
---|---|---|---|---|

−1 | 100 | 0.08 | 0.20 | 0.80 |

0 | 150 | 0.11 | 0.30 | 1.20 |

+1 | 200 | 0.14 | 0.40 | 1.60 |

Mechanical Properties | AISI-52100 | Ceramic |
---|---|---|

Young’s Modulus [GPa] | 210 | 415 |

Density [kg/m^{3}] | 7850 | 3500 |

Poisson’s ratio | 0.30 | 0.22 |

Thermal properties | AISI-52100 | Ceramic |

Heat capacity [J/kgK] | 278 at 93 °C | 334 |

324 at 316 °C | ||

579 at 649 °C | ||

718 at 871 °C | ||

Thermal expansion [μm/mK] | 11.9 | 8.4 |

Thermal conductivity [W/mK] | 24.57 at 149 °C | 7.5 |

24.4 at 349 °C | ||

24.23 at 477 °C | ||

24.75 at 604 °C |

Input | Output | ||||||
---|---|---|---|---|---|---|---|

Standard Order | V_{c} (m/min) | f (mm/rev) | r_{ε} (mm) | a_{p} (mm) | Experimental F_{resultant} (N) | Numerical F_{resultant} (N) | Relative Error (%) |

1 | 100 | 0.08 | 0.80 | 0.25 | 127.1 | 132.8 | 4.5 |

2 | 100 | 0.14 | 0.80 | 0.25 | 187.2 | 203.1 | 8.5 |

3 | 200 | 0.08 | 0.80 | 0.25 | 119.9 | 126.6 | 5.6 |

4 | 200 | 0.14 | 0.80 | 0.25 | 171.7 | 183.3 | 6.8 |

5 | 150 | 0.11 | 1.20 | 0.25 | 141.4 | 139.0 | −1.7 |

6 | 100 | 0.08 | 1.60 | 0.25 | 146.0 | 161.0 | 10.3 |

7 | 100 | 0.14 | 1.60 | 0.25 | 191.6 | 212.4 | 10.8 |

8 | 200 | 0.08 | 1.60 | 0.25 | 128.3 | 126.9 | −1.1 |

9 | 200 | 0.14 | 1.60 | 0.25 | 183.7 | 214.4 | 16.7 |

Input | Output | ||||||
---|---|---|---|---|---|---|---|

Run | V_{c} (m/min) | f (mm/rev) | a_{p} (mm) | r_{ε} (mm) | Numerical F_{resultant} (N) | Statistical F_{resultant} (N) | Relative Error (%) |

1 | 100 | 0.11 | 0.3 | 1.2 | 203.2 | 209.6 | −3.0 |

2 | 150 | 0.11 | 0.3 | 1.2 | 199.8 | 199.1 | 0.4 |

3 | 150 | 0.11 | 0.3 | 1.6 | 210.6 | 199.3 | 5.6 |

4 | 150 | 0.11 | 0.3 | 1.2 | 203.5 | 199.1 | 2.2 |

5 | 150 | 0.11 | 0.4 | 1.2 | 227.7 | 230.8 | −1.3 |

6 | 150 | 0.11 | 0.2 | 1.2 | 127.9 | 120.5 | 6.2 |

7 | 150 | 0.11 | 0.3 | 0.8 | 177.1 | 184.0 | −3.7 |

8 | 200 | 0.11 | 0.3 | 1.2 | 207.3 | 196.5 | 5.5 |

9 | 150 | 0.08 | 0.3 | 1.2 | 217.4 | 195.3 | 11.3 |

10 | 150 | 0.14 | 0.3 | 1.2 | 239.2 | 256.9 | −6.9 |

11 | 200 | 0.14 | 0.2 | 1.6 | 165.4 | 165.5 | 0.1 |

12 | 200 | 0.08 | 0.4 | 1.6 | 206.5 | 216.1 | −4.4 |

13 | 100 | 0.14 | 0.4 | 1.6 | 307.8 | 311.1 | −1.0 |

14 | 200 | 0.14 | 0.4 | 0.8 | 276.6 | 278.8 | −0.8 |

15 | 150 | 0.11 | 0.3 | 1.2 | 200.6 | 199.1 | 0.8 |

16 | 100 | 0.08 | 0.4 | 0.8 | 206.0 | 212.0 | −2.8 |

17 | 100 | 0.08 | 0.2 | 1.6 | 131.4 | 135.3 | −2.9 |

18 | 100 | 0.08 | 0.2 | 0.8 | 114.4 | 119.1 | −3.9 |

19 | 100 | 0.08 | 0.4 | 1.6 | 235.2 | 234.8 | 0.2 |

20 | 150 | 0.11 | 0.3 | 1.2 | 186.6 | 199.1 | −6.3 |

21 | 200 | 0.08 | 0.2 | 0.8 | 108.0 | 110.8 | −2.5 |

22 | 200 | 0.08 | 0.2 | 1.6 | 115.5 | 125.4 | −7.9 |

23 | 200 | 0.08 | 0.4 | 0.8 | 193.8 | 194.7 | −0.5 |

24 | 200 | 0.14 | 0.4 | 1.6 | 295.4 | 293.4 | 0.7 |

25 | 150 | 0.11 | 0.3 | 1.2 | 201.4 | 199.1 | 1.2 |

26 | 200 | 0.14 | 0.2 | 0.8 | 154.5 | 157.6 | −2.0 |

27 | 100 | 0.14 | 0.2 | 0.8 | 168.5 | 165.0 | 2.1 |

28 | 150 | 0.11 | 0.3 | 1.2 | 198.7 | 199.1 | −0.2 |

29 | 100 | 0.14 | 0.2 | 1.6 | 172.5 | 174.3 | −1.0 |

30 | 100 | 0.14 | 0.4 | 0.8 | 302.3 | 295.0 | 2.5 |

Source | Degree of Freedom | Sum of Squares | Mean Square | f-Value | p-Value |
---|---|---|---|---|---|

Model | 15 | 78,097.5 | 5206.5 | 45.63 | 0.000 |

Error | 14 | 1597.4 | 114.1 | ||

Total | 29 | 79,694.9 | |||

R-sq (adj) = 95.85% | |||||

Term | |||||

Blocks | 1 | 76.4 | 76.4 | 0.67 | 0.427 |

V_{c} | 1 | 779.3 | 779.3 | 6.83 | 0.020 |

f | 1 | 17,054.7 | 17,054.7 | 149.47 | 0.000 |

a_{p} | 1 | 54,800.3 | 54,800.3 | 480.28 | 0.000 |

r_{ε} | 1 | 1074.1 | 1074.1 | 9.41 | 0.008 |

V_{c}^{2} | 1 | 40.4 | 40.4 | 0.35 | 0.561 |

f^{2} | 1 | 1857.9 | 1875.9 | 16.28 | 0.001 |

ap^{2} | 1 | 1391.0 | 1391.0 | 12.19 | 0.004 |

r_{ε}^{2} | 1 | 139.5 | 139.5 | 1.22 | 0.287 |

V_{c} × f | 1 | 1.0 | 1.0 | 0.01 | 0.926 |

V_{c} × a_{p} | 1 | 79.2 | 79.2 | 0.69 | 0.419 |

V_{c} × r_{ε} | 1 | 2.0 | 2.0 | 0.02 | 0.896 |

f × a_{p} | 1 | 1389.4 | 1389.4 | 12.18 | 0.004 |

f × r_{ε} | 1 | 46.2 | 46.2 | 0.40 | 0.535 |

a_{p} × r_{ε} | 1 | 45.4 | 45.4 | 0.40 | 0.538 |

Lack of fit | 10 | 1597.4 | 144.7 | 3.86 | 0.102 |

Pure error | 4 | 1447.4 | 37.5 |

Factor | Goal | Lower Limt | Upper Limit |
---|---|---|---|

V_{c} (m/min) | In range | 100 | 200 |

f (mm/rev) | In range | 0.08 | 0.14 |

a_{p} (mm) | In range | 0.20 | 0.40 |

r_{ε} (mm) | In range | 0.80 | 1.60 |

F_{resultant} (N) | Minimize | 108.0 | 307.8 |

Solution | V_{c} (m/min) | f (mm/rev) | a_{p} (mm) | r_{ε} (mm) | F_{resultant} (N) | Desirability |
---|---|---|---|---|---|---|

1 | 175.76 | 0.097 | 0.20 | 0.80 | 101.0 | 1.000 |

2 | 177.78 | 0.098 | 0.20 | 1.20 | 115.0 | 0.965 |

3 | 199.73 | 0.082 | 0.20 | 1.60 | 123.4 | 0.923 |

Test | V_{c} (m/min) | f (mm/rev) | a_{p} (mm) | r_{ε} (mm) | Simulated F _{resultant} (N) | Predicted F _{resultant} (N) | Relative Error (%) |
---|---|---|---|---|---|---|---|

1 | 175.75 | 0.097 | 0.20 | 0.80 | 113.7 | 101.0 | 12.5 |

2 | 120 | 0.10 | 0.25 | 0.80 | 139.5 | 149.6 | −6.8 |

3 | 160 | 0.10 | 0.25 | 1.20 | 141.6 | 159.0 | −10.9 |

4 | 160 | 0.12 | 0.25 | 1.60 | 183.1 | 175.2 | 4.5 |

5 | 120 | 0.10 | 0.35 | 0.80 | 190.5 | 201.4 | −5.4 |

6 | 160 | 0.10 | 0.35 | 1.20 | 229.4 | 210.6 | 8.9 |

7 | 160 | 0.12 | 0.35 | 1.60 | 261.7 | 234.7 | 11.5 |

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**MDPI and ACS Style**

Tzotzis, A.; Tapoglou, N.; Verma, R.K.; Kyratsis, P.
3D-FEM Approach of AISI-52100 Hard Turning: Modelling of Cutting Forces and Cutting Condition Optimization. *Machines* **2022**, *10*, 74.
https://doi.org/10.3390/machines10020074

**AMA Style**

Tzotzis A, Tapoglou N, Verma RK, Kyratsis P.
3D-FEM Approach of AISI-52100 Hard Turning: Modelling of Cutting Forces and Cutting Condition Optimization. *Machines*. 2022; 10(2):74.
https://doi.org/10.3390/machines10020074

**Chicago/Turabian Style**

Tzotzis, Anastasios, Nikolaos Tapoglou, Rajesh Kumar Verma, and Panagiotis Kyratsis.
2022. "3D-FEM Approach of AISI-52100 Hard Turning: Modelling of Cutting Forces and Cutting Condition Optimization" *Machines* 10, no. 2: 74.
https://doi.org/10.3390/machines10020074