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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 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

- -
- 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.
- -
- 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

- Gingold, R.A.; Monaghan, J.J. Smoothed particle hydrodynamics: Theory and application to non-spherical stars. Mon. Not. R. Astron. Soc.
**1977**, 181, 375–389. [Google Scholar] [CrossRef] - Lucy, L.B. A numerical approach to the testing of the fission hypothesis. Astron. J.
**1977**, 82, 1013–1024. [Google Scholar] [CrossRef] - Monaghan, J.; Kos, A.; Issa, N. Fluid motion generated by impact. J. Waterw. Port Coast. Ocean Eng.
**2003**, 129, 250–259. [Google Scholar] [CrossRef] - Afrasiabi, M.; Mohammadi, S. Analysis of bubble pulsations of underwater explosions by the smoothed particle hydrodynamics method. In Proceedings of the ECCOMAS International Conference on Particle Based Methods, Barcelona, Spain, 25–27 November 2009. [Google Scholar]
- Marrone, S.; Colagrossi, A.; Le Touzé, D.; Graziani, G. Fast free-surface detection and level-set function definition in SPH solvers. J. Comput. Phys.
**2010**, 229, 3652–3663. [Google Scholar] [CrossRef] - Afrasiabi, M.; Roethlin, M.; Wegener, K. Thermal simulation in multiphase incompressible flows using coupled meshfree and particle level set methods. Comput. Methods Appl. Mech. Eng.
**2018**, 336, 667–694. [Google Scholar] [CrossRef] - Afrasiabi, M.; Roethlin, M.; Chatzi, E.; Wegener, K. A Robust Particle-Based Solver for Modeling Heat Transfer in Multiphase Flows. In Proceedings of the European Conference on Computational Mechanics and Computational Fluid Dynamics (ECCM-ECFD), Glasgow, UK, 11–15 June 2018. [Google Scholar]
- Afrasiabi, M.; Chatzi, E.; Wegener, K. A Particle Strength Exchange Method for Metal Removal in Laser Drilling. Procedia CIRP
**2018**, 72, 1548–1553. [Google Scholar] [CrossRef] - Afrasiabi, M.; Wegener, K. 3D Thermal Simulation of a Laser Drilling Process with Meshfree Methods. J. Manuf. Mater. Process.
**2020**, 4, 58. [Google Scholar] [CrossRef] - Alshaer, A.; Rogers, B.; Li, L. Smoothed Particle Hydrodynamics (SPH) modelling of transient heat transfer in pulsed laser ablation of Al and associated free-surface problems. Comput. Mater. Sci.
**2017**, 127, 161–179. [Google Scholar] [CrossRef] - Limido, J.; Espinosa, C.; Salaün, M.; Lacome, J.L. SPH method applied to high speed cutting modelling. Int. J. Mech. Sci.
**2007**, 49, 898–908. [Google Scholar] [CrossRef][Green Version] - Villumsen, M.F.; Fauerholdt, T.G. Simulation of metal cutting using smooth particle hydrodynamics. In Proceedings of the German LS-DYNA Forum 2018, Bamberg, Germany, 15–17 October 2008; pp. C-III-17–C-III-36. [Google Scholar]
- Ruttimann, N.; Buhl, S.; Wegener, K. Simulation of single grain cutting using SPH method. J. Mach. Eng.
**2010**, 10, 17–29. [Google Scholar] - Eberhard, P.; Gaugele, T. Simulation of cutting processes using mesh-free Lagrangian particle methods. Comput. Mech.
**2013**, 51, 261–278. [Google Scholar] [CrossRef] - Spreng, F.; Eberhard, P. Machining process simulations with smoothed particle hydrodynamics. Procedia CIRP
**2015**, 31, 94–99. [Google Scholar] [CrossRef][Green Version] - Shaw, M.C. Metal Cutting Principles; Oxford University Press: New York, NY, USA, 2005; Volume 2. [Google Scholar]
- Röthlin, M.; Klippel, H.; Afrasiabi, M.; Wegener, K. Metal cutting simulations using smoothed particle hydrodynamics on the GPU. Int. J. Adv. Manuf. Technol.
**2019**, 102, 3445–3457. [Google Scholar] [CrossRef] - Roethlin, M.; Klippel, H.; Afrasiabi, M.; Wegener, K. Meshless single grain cutting simulations on the GPU. Int. J. Mechatron. Manuf. Syst.
**2019**, 12, 272–297. [Google Scholar] [CrossRef] - Afrasiabi, M.; Meier, L.; Röthlin, M.; Klippel, H.; Wegener, K. GPU-accelerated meshfree simulations for parameter identification of a friction model in metal machining. Int. J. Mech. Sci.
**2020**, 176, 105571. [Google Scholar] [CrossRef] - Afrasiabi, M.; Roethlin, M.; Klippel, H.; Wegener, K. Meshfree simulation of metal cutting: An updated Lagrangian approach with dynamic refinement. Int. J. Mech. Sci.
**2019**, 160, 451–466. [Google Scholar] [CrossRef] - Fatehi, R.; Manzari, M. Error estimation in smoothed particle hydrodynamics and a new scheme for second derivatives. Comput. Math. Appl.
**2011**, 61, 482–498. [Google Scholar] [CrossRef] - Afrasiabi, M.; Roethlin, M.; Wegener, K. Contemporary Meshfree Methods for Three Dimensional Heat Conduction Problems. Arch. Comput. Methods Eng.
**2020**, 27, 1413–1447. [Google Scholar] [CrossRef] - Issa, M.; Saanouni, K.; Labergère, C.; Rassineux, A. Prediction of serrated chip formation in orthogonal metal cutting by advanced adaptive 2D numerical methodology. Int. J. Mach. Mach. Mater.
**2011**, 9, 295–315. [Google Scholar] [CrossRef] - Taylor, G.I.; Quinney, H. The latent energy remaining in a metal after cold working. Proc. R. Soc. Lond. A
**1934**, 143, 307–326. [Google Scholar] - Johnson, G.R. A constitutive model and data for materials subjected to large strains, high strain rates, and high temperatures. In Proceedings of the Seventh International Symposium on Ballistics, The Hague, The Netherlands, 19–21 April 1983; pp. 541–547. [Google Scholar]
- Özel, T. The influence of friction models on finite element simulations of machining. Int. J. Mach. Tools Manuf.
**2006**, 46, 518–530. [Google Scholar] [CrossRef] - Liu, M.; Liu, G. Smoothed particle hydrodynamics (SPH): An overview and recent developments. Arch. Comput. Methods Eng.
**2010**, 17, 25–76. [Google Scholar] [CrossRef][Green Version] - Monaghan, J.J. Smoothed particle hydrodynamics. Rep. Prog. Phys.
**2005**, 68, 1703. [Google Scholar] [CrossRef] - Monaghan, J.J. Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys.
**1992**, 30, 543–574. [Google Scholar] [CrossRef] - Brookshaw, L. A method of calculating radiative heat diffusion in particle simulations. Proc. Astron. Soc. Aust.
**1985**, 6, 207–210. [Google Scholar] [CrossRef] - Deligonul, Z.; Bilgen, S. Solution of the Volterra equation of renewal theory with the Galerkin technique using cubic splines. J. Stat. Comput. Simul.
**1984**, 20, 37–45. [Google Scholar] [CrossRef] - Monaghan, J.; Gingold, R. Shock simulation by the particle method SPH. J. Comput. Phys.
**1983**, 52, 374–389. [Google Scholar] [CrossRef] - Gray, J.; Monaghan, J.; Swift, R. SPH elastic dynamics. Comput. Methods Appl. Mech. Eng.
**2001**, 190, 6641–6662. [Google Scholar] [CrossRef] - Monaghan, J. On the problem of penetration in particle methods. J. Comput. Phys.
**1989**, 82, 1–15. [Google Scholar] [CrossRef] - Afrasiabi, M. Thermomechanical Simulation of Manufacturing Processes Using GPU-Accelerated Particle Methods. Ph.D. Thesis, ETH Zurich, Zurich, Switzerland, 2020. [Google Scholar] [CrossRef]
- Randles, P.; Libersky, L. Smoothed particle hydrodynamics: Some recent improvements and applications. Comput. Methods Appl. Mech. Eng.
**1996**, 139, 375–408. [Google Scholar] [CrossRef] - Arrazola, P.; Özel, T.; Umbrello, D.; Davies, M.; Jawahir, I. Recent advances in modelling of metal machining processes. CIRP Ann.-Manuf. Technol.
**2013**, 62, 695–718. [Google Scholar] [CrossRef] - Stöcker, H. Taschenbuch der Physik; In Deutsch: Zurich, Switzerland, 2000. [Google Scholar]
- Niu, W.; Mo, R.; Liu, G.; Sun, H.; Dong, X.; Wang, G. Modeling of orthogonal cutting process of A2024-T351 with an improved SPH method. Int. J. Adv. Manuf. Technol.
**2018**, 95, 905–919. [Google Scholar] [CrossRef] - Ducobu, F.; Rivière-Lorphèvre, E.; Filippi, E. On the importance of the choice of the parameters of the Johnson-Cook constitutive model and their influence on the results of a Ti6Al4V orthogonal cutting model. Int. J. Mech. Sci.
**2017**, 122, 143–155. [Google Scholar] [CrossRef] - Saelzer, J.; Berger, S.; Iovkov, I.; Zabel, A.; Biermann, D. In-situ measurement of rake face temperatures in orthogonal cutting. CIRP Ann.
**2020**, 69, 61–64. [Google Scholar] [CrossRef] - Sima, M.; Özel, T. Modified material constitutive models for serrated chip formation simulations and experimental validation in machining of titanium alloy Ti–6Al–4V. Int. J. Mach. Tools Manuf.
**2010**, 50, 943–960. [Google Scholar] [CrossRef]

**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${}^{-4}$ | $5.67\times {10}^{-8}$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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