# Smoothed Particle Hydrodynamics Simulation of Orthogonal Cutting with Enhanced Thermal Modeling

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## Abstract

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## 1. Introduction

## 2. Governing Equations

## 3. SPH Formulation

## 4. Proposed Thermal Model

#### 4.1. Discretization of Laplacian

#### 4.2. Thermal Boundary Conditions

## 5. Results & Discussion

#### 5.1. Preliminary Study

#### 5.2. Application: Orthogonal Metal Cutting

#### 5.2.1. Force Prediction and Chip Shape

^{®}i5-4690 at 3.50 GHz. Furthermore, the orthogonal cutting experiment was conducted by setting a rake angle of $\gamma =0$° in order to obtain a state, where the thrust force (${F}_{t}$ in Figure 4) is mainly caused by the friction between the rake face and the sliding chip.

#### 5.2.2. Temperature Distribution

#### 5.2.3. Effect of SPH Particles Size

## 6. Conclusions

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- To account for the heat loss boundary condition in SPH cutting models, a new approach was proposed and subsequently applied to an orthogonal cutting problem. This approach incorporates an efficient surface detection algorithm, which was initially proposed for fluid flow applications [5]. An improved thermal model representing more realistic boundary conditions was presented as a result of this development. In addition to the constant temperature at fixed boundaries, heat loss in the forms of thermal convection and radiation was included.
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- To ensure a second-order Laplacian operator in SPH, a corrected formulation (see the FMFS scheme in Equation (13)) was implemented for the first time in a metal cutting problem. This higher-order method allows us to solve the heat equation more accurately, especially in the presence of free surfaces.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Illustration of the proposed approach for thermal boundary conditions. Left: introduction of the surface detection algorithm into SPH cutting models. The implementation considers the tool (rigid) and the workpiece (deformable) as two separate bodies. Right: only particles with $\lambda <0.75$ are included and then categorized into 2 surface groups (red and blue). Thermal boundary conditions are imposed on these blue and red surface particles accordingly.

**Figure 6.**Summary of the measured and predicted forces for the cutting test. Forces predicted by SPH with different coefficients of friction are plotted in the bar chart, where the proposed thermal model is utilized. Impact of $\mu $ on the thrust forces is higher than the cutting forces, demonstrating the most accurate result when $\mu =0.65$ is used.

**Figure 7.**Distribution of the temperature after 2 mm of cut. Lower temperatures are observed in the enhanced model by comparing the two black frames. Simulated chip shapes are superimposed to give insights into the impact of the heat loss boundary condition on chip curling.

**Figure 8.**Distributions and contours of temperature obtained by SPH using the reference and proposed thermal model. An average of the rake face temperature in simulation results (considering the particles inside the black frames) is compared to the experimental measurement provided by [41]. The color bar is limited to 945 K for better visibility.

**Figure 9.**Effect of SPH particles size on the chip shape, where $\Delta x$ is the diameter of SPH particles.

**Figure 10.**Comparison of chip shapes in different SPH resolutions, where ${v}_{c}=241$ m/min, ${t}_{u}=0.1$ mm, and ${r}_{\beta}=0.005$ mm. The experimental photograph reprinted from Sima et al. [42] Copyright (2021), with permission from Elsevier under License Number 4991260002261.

Body | Property | Symbol | Unit | Value |
---|---|---|---|---|

Tool | Clearance angle | $\alpha $ | deg | 7 |

Rake angle | $\gamma $ | deg | 0 | |

Cutting edge radius | ${r}_{\beta}$ | mm | 0.0028 | |

Speed | ${v}_{c}$ | m min${}^{-1}$ | 318.5 | |

Heat conductivity | k | W m${}^{-1}$ K${}^{-1}$ | 88 | |

Specific heat capacity | ${c}_{p}$ | J kg${}^{-1}$ K${}^{-1}$ | 292 | |

Workpiece | Length | ${l}_{x}$ | mm | 3 |

Height | ${l}_{y}$ | mm | 0.3 | |

Uncut chip thickness | ${t}_{u}$ | mm | 0.1 | |

Cut distance | ${l}_{c}$ | mm | 2 | |

Density | $\rho $ | kg m${}^{-3}$ | 4430 | |

Young’s modulus | E | GPa | 113.8 | |

Poisson ratio | $\nu $ | – | 0.35 | |

Heat conductivity | k | W m${}^{-1}$ K${}^{-1}$ | 7.3 | |

Specific heat capacity | ${c}_{p}$ | J kg${}^{-1}$ K${}^{-1}$ | 580 | |

Reference temperature | ${T}_{r}$ | K | 300 | |

Melting temperature | ${T}_{m}$ | K | 1878 | |

Coeff. of friction | $\mu $ | – | 0.35 | |

Pct. of plastic work into heat | $\chi $ | – | 90% | |

Pct. of frictional work into heat | $\eta $ | – | 100% | |

All | Convection coeff. | ${h}_{c}$ | W m${}^{-2}$ K${}^{-1}$ | 50.0 |

Emissivity coeff. | $\u03f5$ | – | 0.30 | |

Stefan–Boltzmann constant | $\sigma $ | W m${}^{-2}$ K${}^{-4}$ | $5.67\times {10}^{-8}$ |

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**MDPI and ACS Style**

Afrasiabi, M.; Klippel, H.; Roethlin, M.; Wegener, K.
Smoothed Particle Hydrodynamics Simulation of Orthogonal Cutting with Enhanced Thermal Modeling. *Appl. Sci.* **2021**, *11*, 1020.
https://doi.org/10.3390/app11031020

**AMA Style**

Afrasiabi M, Klippel H, Roethlin M, Wegener K.
Smoothed Particle Hydrodynamics Simulation of Orthogonal Cutting with Enhanced Thermal Modeling. *Applied Sciences*. 2021; 11(3):1020.
https://doi.org/10.3390/app11031020

**Chicago/Turabian Style**

Afrasiabi, Mohamadreza, Hagen Klippel, Matthias Roethlin, and Konrad Wegener.
2021. "Smoothed Particle Hydrodynamics Simulation of Orthogonal Cutting with Enhanced Thermal Modeling" *Applied Sciences* 11, no. 3: 1020.
https://doi.org/10.3390/app11031020