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In this work, investigation of particle rebound characteristics due to impact with surface of a target material is presented. The rebound of a spherical particle after impact on a planar surface was analyzed in detail. Specifically, the coefficient of restitution of the particle under various impact conditions was investigated numerically. This study has been conducted by carrying out a series of FEM-based (finite element method) simulations using ANSYS Autodyn software. First, a summary about the state of the art and the theoretical models for the elastic collisions were reviewed. Afterwards, the impact of an aluminum oxide particle on an aluminum alloy target surface was modeled. Using the Autodyn tool, the results were compared and validated by the experimental results of Gorham and Kharaz [1]. Selection of an appropriate equation of state (EOS) and a strength model for each material had a strong effect on the results. For both materials, the Shock EOS was applied for the final simulations. As the strength model, the Johnson-Cook and the elastic model were used, respectively. The agreement of the obtained numerical results with the experimental data confirmed that the proposed model can precisely predict the real behavior of the particle after the impact, when the material models are properly chosen. Furthermore, the effects of impact velocity and impact angle on the rebound characteristics of the particle were analyzed in detail. It was found that the selection of the exact value of friction coefficient has a drastic effect on the prediction of restitution coefficient values, especially the tangential restitution coefficient.

The coefficient of restitution of a particle impacting a target surface is the ratio of the particle velocities after and before the impact. This positive real number varies from 0.0 to 1.0 depending on the characteristics of the materials used and the impact conditions. The coefficient of restitution e, for a normal impact as depicted in

e = | V r | | V i | (1)

where V_{r} is the rebound velocity and V_{i} is the impact velocity. During normal impacts, where no tangential component is included between the particle and the target surface, there are two ways of losing energy. One way is due to the dissipation of the stress wave propagation and the other one due to the plastic deformation. In the case of a colliding sphere, three different kinds of deformation can be identified; elastic, elastic-plastic and fully plastic. An elastic shock takes place at the beginning of the collision. When the stress is large enough, plastic deformation begins to occur in the area of the contact point. This deformation depends mainly on the impact velocity and on the material properties of both the materials involved, i.e. the particle and target. When the maximum compression is achieved, an amount of elastic energy is stored. At this moment, the contact force starts to decrease by releasing some of the energy back to the sphere and promoting the rebound of the particle. This restitution phase can be considered as elastic because only elastic strain energy is released. If an impact is perfectly elastic, energy is dissipated by the propagation of elastic waves. Hunter (1957) [

When analysing elastic impacts, the behaviour of such collisions can be introduced theoretically using Hertz theory (1881) through a quasi-static procedure

[

e = V y V i [ 8 5 ( V i V y ) 2 − 3 5 ] 3 / 8 (2)

where V_{y} is the yield point velocity. For V_{i} = V_{y}, e = 1.

For V i ≫ V y , the Equation (2) simplifies to:

e = 1.193 ( V y V i ) 1 / 4 (3)

However, Stronge’s Equation (2) predicts e > 1 for V y < V i < 1.59 V y , which is an impossible condition. Thornton (1997) [

e 2 = 1.442 [ 1 − 1 6 ( V y V i ) 2 ] 1 / 2 × [ ( V y V i ) ( V y V i ) + 2 6 5 − 1 5 ( V y V i ) 2 ] 1 / 4 (4)

It satisfies the condition e = 1.0 when Vi = V_{y}. At high velocities, (V_{y}/V_{i})^{2} → 0 and Equation (4) becomes:

e = ( 6 3 2 ) 1 / 2 [ V y V y + 2 6 5 V i ] 1 / 4 (5)

And finally if V i ≫ V y , the following relation for coefficient e is obtained:

e = 1.185 ( V y V i ) 1 / 4 (6)

In the case analysed in Thornton’s work [_{y}, is defined as:

V y = ( π 2 E * ) 2 ( 2 5 ρ ) 1 / 2 σ y 5 / 2 = 1.56 ( σ y 5 E * 4 ρ ) 1 / 2 (7)

For this particular case, by substituting Equation (7) into Equation (6), the following expression is obtained:

e = 1.324 ( σ y 5 E * 4 ρ ) 1 / 8 ( V i ) − 1 / 4 (8)

Later, Vu-Quo and Zhang (1999) [

Jackson and Green (2010) [_{c}. Furthermore, the effects of varying material and geometrical properties like the yield strength, elastic modulus, Poisson’s ratio and sphere radius were investigated [

The FEM should be understood as a method for finding an approximate solution for a simplified model. Numerical treatment reduces the simplified model to a form which is solvable by a finite number of numerical operations. This means that the approximate solution has to be characterized by a finite number of parameters, called degrees of freedom. One of the earliest work in the field of particle impact that used a finite element software was the work done by Lim and Strong (1998) [

Vu-Quo and Zhang [

Zhang and Vu-Quo (2001) [

Wu et al. [

e = 0.78 ( V i / V y E * / Y ) − 1 / 2 (9)

where E^{*} is the normalized elastic modulus and Y is the yield stress. And for the impacts of an elastic-perfectly plastic sphere with a rigid wall:

e = 0.62 ( V i / V y E * / Y ) − 1 / 2 (10)

The onset of finite plastic-deformation impact has been determined in terms of V_{i}/V_{y} and E^{*}/Y for different impact configurations. The FEM results showed that for the impact of small deformation, the coefficient of restitution is mainly dependant on V_{i}/V_{y}, which is consistent with those predicted by the theory of impact mechanics. While for the impact of finite plastic deformation, the restitution coefficient is also dependent on E^{*}/Y. The conclusion of their work was that when finite plastic deformation occurs, the restitution coefficient is proportional to [ ( V i / V y ) / ( E * / Y ) ] − 1 / 2 .

In another work, Thornton et al. [

e t = 1 − μ ( 1 + e n ) / tan θ i (11)

where μ is the Coulomb coefficient of friction and θ i is the particle impact angle.

Thornton et al. [

In this work, investigation of particle rebound characteristics due to impact with surface of a target material is presented. The rebound of a spherical particle after impact on a planar surface was analyzed in detail. Specifically, the coefficient of restitution of the particle under various impact conditions was investigated numerically. This study has been conducted by carrying out a series of finite element method (FEM) based simulations using ANSYS Autodyn software.

Experimental measurement of the rebound particle characteristics with an extremely good precision is a time-consuming process. However, this is necessary for validation of numerical results. In this part of the work, the measurement technique used in works of Gorham and Kharaz [

99.5% Aluminium oxide (particle) | Aluminium alloy 2024 (target material) | ||||
---|---|---|---|---|---|

Mechanical properties | Units of measure | Value | Value | ||

Density | kg/m^{3} | 3890 | 2780 | ||

Elastic modulus | GPa | 360 | 70 | ||

Shear modulus | GPa | 154 | 26 | ||

Bulk modulus | GPa | 228 | - | ||

Flexural strength | MPa | 379 | - | ||

Poisson’s ratio | - | 0.23 | 0.33 | ||

Compressive strength | MPa | 2600 | - | ||

Hardness | kg/mm^{2} | 1440 | 47 | ||

Elongation at break | - | - | 19% | ||

Yield strength | MPa | - | 324 | ||

Thermal properties | |||||

Thermal conductivity | W/m・K | 35 | 121 | ||

Coefficient of thermal expansion | 10^{−}^{6}/C | 8.4 | 24.66 | ||

Specific heat | J/kg・K | 880 | 875 | ||

The geometries of the 5 mm spherical particle and the anvil with a size of 140 × 125 × 25 mm were generated. In the studied case, the longitudinal wave speed in the target material is around 5.3 km/s [

1) Equation of State (EOS)

The hydrodynamic response of a material is described by the equation of state. This is the primary response for gases and liquids, which can withstand no shear. Their response to dynamic loading is assumed to be hydrodynamic, with pressure varying as a function of internal energy and density. However, for the solid phase, this is also the primary response at high deformation rates, when the hydrodynamic pressure is far greater than the yield stress of the solid material [_{p} (particle velocity) and U (shock propagation velocity). The choice of this equation of state has been proved in other studies (e.g. by Corbett [

U = c 0 + s u p (12)

where s is the linear Hugoniot slope coefficient and c_{0} is the sound speed in the material. This equation of state is specified in the Mie and Gruneisen form [

p = p H + Γ ρ ( e − e H ) (13)

where it is assumed that Γ ρ = Γ 0 ρ 0 is constant and the pressure and energy terms are defined as Equation (14) and Equation (15), respectively.

p H = ρ 0 c 0 2 μ ( 1 − μ ) ( 1 − ( s − 1 ) μ ) 2 (14)

e H = 1 2 ρ H ρ 0 ( μ 1 + μ ) (15)

where Γ 0 is the Gruneisen parameter, µ = ( ρ / ρ 0 ) − 1 , ρ is the current density and ρ_{0} is the initial density. The equation of state in this form is only valid for a limited impact velocity range, because it does not include a phase change, such as melting or vaporization.

2) Material Strength Model

Solid materials may initially respond elastically, but under highly dynamic loadings, they can reach stress states that exceed their yield stress and deform plastically. Material strength laws describe this nonlinear elastic-plastic response. In this study, Johnson-Cook strength model [

σ = ( A + B ε n ) ( 1 + C ln ε ˙ ∗ ) ( 1 − T ∗ m ) (16)

where σ is the flow stress, σ_{0} is the yield strength, ε is the equivalent plastic strain, ε ˙ * is the normalized plastic strain rate and T^{*} is the normalized temperature based on a reference melt temperature. The five material parameters are σ_{0}, B, C, n and m. The expression in the first bracket is the stress as a function of elongation when ε ˙ * = 1.0 s − 1 ^{ }and T^{*} = 0. The constants B and n represent the effect of cold deformation. The expression in the second bracket represents the effect that the elongation rate has on the yield strength of the material. The last term represents the thermal softening so that the yield strength approaches zero when the melting temperature is reached. The constant m represents the thermal softening exponent.

As it is explained in the state of the art, the experimental data of Gorham and Kharaz [

As it can be observed in

As it can be observed in

Spherical particle | |
---|---|

Material used: | Al_{2}O_{3} 99.5% |

Equation of state: | Shock |

Strength model: | Von Mises [ |

Target material | |

Material used: | Al 2024 |

Equation of state: | Shock |

Strength model: | None |

Spherical particle | |
---|---|

Material used: | Al_{2}O_{3} 99.5% |

Equation of state: | Polynomial |

Strength model: | Johnson-Holmquist [ |

Target material | |

Material used: | Al 2024 |

Equation of state: | Shock |

Strength model: | Johnson-Cook |

the material, the next step was to modify the mesh with a better precision. A third body has been created inside the big anvil. This body has been located exactly where the impact takes place as shown in

Spherical particle | |
---|---|

Material used: | Al_{2}O_{3} 99.5% |

Equation of state: | Shock |

Strength model: | Elastic |

Target material | |

Material used: | Al 2024 |

Equation of state: | Shock |

Strength model: | Johnson-Cook |

different meshes were studied. The sphere and the part of the surface where the collision takes place were meshed with an equal element size. In the other part of the surface, where the stress is negligible, the element size can be generated coarser. The mesh has been refined step by step. The number of elements was increased by decreasing the element size.

In

Grid | Elements nr. | Restitution coeff. | Deviation |
---|---|---|---|

1 | 6865 | 1.00 | ---- |

2 | 27,909 | 1.00 | ---- |

3 | 70,369 | 1.00 | ---- |

4 | 121,610 | 0.80 | 20% |

5 | 200,765 | 0.723 | 9.6% |

6 | 315,562 | 0.708 | 2.1% |

7 | 475,401 | 0.698 | 1.4% |

8 | 618,240 | 0.696 | 0.2% |

In

In the previous part of the work, all the simulations that have been presented were collision cases with only a normal velocity component. In this part, several simulations with also a tangential component of impact were performed.

Gorham and Kharaz [

they covered a collision angle ranging from 5˚ to 88˚. In the following diagrams, the obtained simulation results are validated with the experimental data of Gorham and Kharaz [

when the impact angle approaches 90˚.

Although the trends of the experimental data and simulation values are similar in _{n}, on the impact angle θ_{i}, and on the friction coefficient µ, based on Equation (11).

In

The challenge to find a theoretical model to predict the rebound behaviour of a particle when some plastic deformation takes place was observed. During the past decades, several theoretical models have been presented that tried to predict the rebound of the particle. However, the theoretical models predict the trend qualitatively good but not quantitatively well. In the present study, the analysed

impact velocities were between 0.5 and 7 m/s and plasticity started in the aluminium alloy surface from the impact velocity of about 0.7 m/s. Therefore, almost the whole investigation occurred in an elasto-plastic regime.

Despite the difficulty of predicting the restitution coefficient for collisions with plastic deformation, by applying the finite element method, ANSYS Autodyn tool, the obtained results had a good agreement with the experimental values [_{t} did not suffer big variations, however e_{n} decreased notably. By an increase of the impact angle from 0˚ to 90˚, the tangential restitution coefficient decreases almost linearly and reaches a minimum value and increases gradually again when the impact angle approaches closer to the 90˚. The normal restitution coefficient decreased asymptotically by an increase of the impact angle.

During the evaluation of the work, it has been observed that the final mesh chosen for the simulations had a big importance to accurately predict the impact. Furthermore, in the present study, the shock equation of state provided better results and was used for both aluminium oxide and aluminium alloy 2024. The strength model that provided the best response was the Johnson-Cook model for the target, and the elastic model for the impacting spherical particle. Selection of the correct friction coefficient had an important influence on the precise prediction of tangential coefficient of restitution and therefore on the rebound velocity and rebound angle of the particle.

The authors would like to thank “Stiftung Rheinland-Pfalz für Innovation, Mainz, Germany” for financial support.

Azimian, M. and Bart, H.-J. (2017) Numerical Investigation of Particle Rebound Characteristics with Finite Element Method. Open Journal of Fluid Dynamics, 7, 310-329. https://doi.org/10.4236/ojfd.2017.73020