# Hybrid Smoothed-Particle Hydrodynamics/Finite Element Method Simulation of Water Droplet Erosion on Ductile Metallic Targets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Water Droplet Erosion Mechanisms

## 3. Modeling Methods

#### 3.1. Johnson–Cook Material Constitutive Model for the Target

#### 3.2. Johnson–Cook Failure Model for the Target

#### 3.3. Smoothed-Particle Hydrodynamics (SPH) Method for Water Droplets

## 4. Numerical Simulations of Multiple Droplet Impacts

#### 4.1. Finite Element Numerical Simulation of the Target Surface

#### 4.2. Smoothed-Particle Hydrodynamics Simulation of the Water Droplets

## 5. Results and Discussion

^{3}, occurs only for angles above 60 degrees. At that specific angle, element removal occurs only after roughly 650 impacts, which could be considered the end of the incubation regime (designated as the “eir” variable for each angle in the figure). The erosion is much more effective for an impacting angle of 70°. At angles of 80° and 90°, the onset and evolution of the erosion regime show similar features. Figure 3b shows the total volume loss after 1000 impacts at a velocity of 600 m/s for the whole range of impact angles.

^{3}, whereas the corresponding volume loss for v = 1000 m/s is 211.172 mm

^{3}(approximately 10 times larger). This is due to the strong non-linear dynamics of the erosion process and the complex mechanisms that emerge when impact velocity and angle are varied.

## 6. Conclusions

- For impact velocities below certain threshold, erosion eventually ceases to occur (within a reasonable lifetime of the metallic targets and independent of any impact angle).
- Above the threshold velocity mentioned before, an incubation regime exists where neither mass removal nor surface roughening occurs. The duration of the incubation regime and the onset of erosion (in terms of the number of accumulated impacts) decreases with increasing impact velocity/kinetic energy.
- Depending on the impact velocity, there is an impact angle lower than 90° where the volume loss is maximum. Namely, in our simulations, the maximum is 80° for velocities v = 600 m/s and 800 m/s and 70° for 1000 m/s
- In spite of these achievements, it is important to emphasize that the model used in this work is a simplification of the physical mechanisms taking place during real water droplet erosion. Moreover, the specific ranges of simulation parameters selected in this work, namely large water droplet sizes (4 mm diameter) and high velocities (600–1000 m/s), allowed us to obtain erosion using reasonable computational resources. However, for this very reason, it is difficult to conduct a direct quantitative comparison with experiments because there are currently no data available on the duration of the incubation period for the combination of ranges of droplet sizes and impact velocities used in this study (to the best of our knowledge).
- Further work should focus on applying more sophisticated material and damage models, taking into account the influence of temperature and corrosion among other variables. Moreover, more realistic combinations of droplet size and impact velocity ranges should be used in order to compare with the experimental results of the literature. A second goal would be to decrease the size of the finite elements and increase the number of SPH particles representing the water droplet. Finally, it is worth considering the application of an SPH model (with the relevant material and damage model) to the impacted metallic surface itself in order to produce a more realistic evolution of the extreme deformations that occur on the real surfaces of material alloys.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Generation of Random Impacts

**Figure A1.**(

**a**) Five water droplets (represented by a spherical volume of SPH particles with a 4 mm diameter) almost simultaneously impact the FEM simulation box representing the metallic target surface. (

**b**) The top view of the xy-plane. In order to avoid damage or removal of finite elements of the “outer” part, random impacts occur only on a smaller area of the “inner” part shown by a red square. Magenta horizontal lines show the eventual maximum range of eventual damage/element removal. (

**c**) A view of the xz-plane of the same five impacting particles. The dashed lines indicate the direction of the impacts (impact angle of 30°). See a detailed description in the text.

**Figure A2.**Three temporal frames of an individual SPH/FEM simulation of the impact of five water droplets at the same velocity of 1000 m/s and impact angle of 70°on random positions as described in Figure A1. Please note that only the “inner part” of the target is being depicted here (dimensions 18 × 18 × 5 mm, see Figure 2a). (

**a**) At t = 0 s, the simulation starts loading the target with the deformations, stress, damage, and removed elements of previous impacts and a new set of five droplets moving toward the surface. (

**b**) At t = 2.9 ms, the five droplets are interacting with the eroded target. (

**c**) Finally, at t = 8.7 ms, the SPH particles representing the droplet rebound off the surface. The simulation of the next five droplets starts with this final state target FEM mesh.

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**Figure 1.**A schematic representation of the physical processes modeled. (

**a**) The droplet impact on an initially flat and intact surface produces “water hammer” pressure and water jetting (in perpendicular direction as indicated by the vertical blue arrow). (

**b**) After a number of impacts, internal stresses, damage, and plastic deformation lead to dimple formation. (

**c**) Repeated water jetting shears the rims of the dimples and their related asperities, which adds to the accumulated damage and fracture. (

**d**) At the advanced stage of the incubation period, the surface is roughened, thus altering the effectivity of the droplet impacts. See a detailed description in the text.

**Figure 2.**(

**a**) The finite element simulation box geometry of the metallic target. (

**b**) The target is composed of the “inner” part (green box) and “outer” part (gray volume) with the shown dimensions. The inner and outer parts are tied together by means of a “tie” constraint. The simulated droplets only impact the “inner” part. See a detailed description in the text.

**Figure 3.**(

**a**) The volume loss measured in mm

^{3}versus the accumulated number of 4 mm diameter water droplet impacts for impact velocity $v=600$ m/s at different impact angles (for angles lower than 60° there is no erosion. The variable “eir” (from “end of incubation regime”) is approximately the number of impacts required for the erosion for each impact angle. (

**b**) The total volume loss after 1000 impacts for different impact angles. See a detailed description in the text.

**Figure 4.**(

**a**) The volume loss measured in mm

^{3}versus the accumulated number of water droplet impacts for impact velocity $v=800$ m/s at different impact angles. The variable “eir” (from “end of incubation regime”) is approximately the number of impacts required for erosion for each impact angle. For angles lower than 40°, there is no erosion. (

**b**) The total volume loss after 1000 impacts for different impact angles. See a detailed description in the text.

**Figure 5.**(

**a**) The volume loss measured in mm

^{3}versus the accumulated number of water droplet impacts for impact velocity $v=1000$ m/s at different impact angles. The variable “eir” (from “end of incubation regime”) is approximately the number of impacts required for erosion for each impact angle. For angles lower than 30°, there is no erosion. (

**b**) The total volume loss after 1000 impacts for different impact angles. See a detailed description in the text.

**Table 1.**The Ti6Al4V composition according to ASTM standards [40].

Component | Al | Fe | O | Ti | V |
---|---|---|---|---|---|

Weight (%) | 6 | 0.25 | 0.2 | 10 | 4 |

**Table 2.**The Ti6Al4V mechanical properties according to ASTM standards [40].

Density (kg/m^{3}) | 4420 |

Modulus of elasticity (GPa) | 115 |

Poisson’s Ratio | 0.32 |

Ultimate tensile strength (MPa) | 950 |

**Table 3.**The Johnson–Cook constitutive material model parameters of the Ti6A4V alloy considered in this work.

Parameter | Value ^{1} |
---|---|

$A$ (MPa) | 1098 |

B (MPa) | 1092 |

$c$ | 0.014 |

$n$ | 0.93 |

$m$ | 1.1 |

$T$(K) | 300 |

${T}_{f}$(K) | 1878 |

${T}_{0}$(K) | 293 |

^{1}According to [41].

${\mathit{d}}_{1}$ | ${\mathit{d}}_{2}$ | ${\mathit{d}}_{3}$ | ${\mathit{d}}_{4}$ | ${\mathit{d}}_{5}$ |
---|---|---|---|---|

−0.09 | 0.27 | 0.48 | 0.014 | 3.870 |

**Table 5.**The Mie–Grüneisen equation of state parameters of Equation (9) used in this work for the SPH modeling of impacting droplets.

${\mathit{\rho}}_{0}$ | ${\mathit{c}}_{0}$ | $\mathit{s}$ | ${\mathsf{\Gamma}}_{0}$ |
---|---|---|---|

1000 Kg/m^{3} | 1482 m/s | 0 | 0 |

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**MDPI and ACS Style**

Mora, A.; Xu, R.; Schmauder, S.
Hybrid Smoothed-Particle Hydrodynamics/Finite Element Method Simulation of Water Droplet Erosion on Ductile Metallic Targets. *Metals* **2023**, *13*, 1937.
https://doi.org/10.3390/met13121937

**AMA Style**

Mora A, Xu R, Schmauder S.
Hybrid Smoothed-Particle Hydrodynamics/Finite Element Method Simulation of Water Droplet Erosion on Ductile Metallic Targets. *Metals*. 2023; 13(12):1937.
https://doi.org/10.3390/met13121937

**Chicago/Turabian Style**

Mora, Alejandro, Ruihan Xu, and Siegfried Schmauder.
2023. "Hybrid Smoothed-Particle Hydrodynamics/Finite Element Method Simulation of Water Droplet Erosion on Ductile Metallic Targets" *Metals* 13, no. 12: 1937.
https://doi.org/10.3390/met13121937