# A linear model of elasto-plastic and adhesive contact deformation

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## Abstract

Rigorous non-linear models of elasto-plastic contact deformation are time-consuming in numerical calculations for the distinct element method (DEM) and quite often unnecessary to represent the actual contact deformation of common particulate systems. In this work a simple linear elasto-plastic and adhesive contact model for spherical particles is proposed. Plastic deformation of contacts during loading and elastic unloading, accompanied by adhesion are considered, for which the pull-off force increases with plastic deformation. Considering the collision of a spherical cohesive body with a rigid flat target, the critical sticking velocity and coefficient of restitution in the proposed model are found to be very similar to those of Thornton and Ning’s model. Sensitivity analyses of the model parameters such as plastic, elastic, plastic-adhesive stiffnesses and pull-off force on work of compaction are carried out. It is found that by increasing the ratio of elastic to plastic stiffness, the plastic component of the total work increases and the elastic component decreases. By increasing the interface energy, the plastic work increases, but the elastic work does not change. The model can be used to efficiently represent the force-displacement of a wide range of particles, thus enabling fast numerical simulations of particle assemblies by the DEM.

### Keywords

Distinct element model (DEM) Contact force model Cohesive powder Plastic deformation Coefficient of restitution Normal impact Adhesion## 1 Introduction

The macroscopic bulk behaviour of powders is governed by the microscopic activity of the individual particles in an assembly. This implies that in order to gain a better understanding of particulate systems and their functioning, the particle interactions at the microscopic level must be analysed. It is currently very difficult to investigate the behaviour of individual particles within a bulk assembly experimentally. Therefore it is helpful to model the behaviour of particles by the use of numerical simulations. Furthermore the use of computer simulations provides a cost effective method as an alternative to experiments, since no physical material or process equipment is required, provided the simulation results are validated. Simulations are also invaluable for cases for which actual experiments are hazardous. For particulate solids, the most appropriate approach for this purpose is the use of computer simulation by the distinct element method (DEM). More details on the methodology of the DEM and its applications are presented elsewhere [1, 2]. Various contact models have been proposed in the literature for evaluation of forces arising from inter-particle collisional interactions. Ning [3] categorized the factors involved in a single contact into three main categories: impact parameters such as particle velocity and impact angle, particle properties such as size, shape, density, surface friction, adhesion and roughness, and environmental factors such as temperature. Considering such a wide range of factors involved in the interactions, modelling of inter-particle contacts is a particularly complex process. Various contact models have been developed and reported in the literature for elastic [1, 4, 5, 6], elastic-adhesive [7, 8, 9, 10, 11, 12, 13], elasto-plastic [14, 15, 16, 17] and elasto-plastic and adhesive [18, 19, 20, 21] contacts, some of which involve complex mathematical equations. The more complex a model, the slower the simulations. Simplifications can be done in order to reduce the computational complexity of the models; however this comes at the expense of losing the accuracy of capturing the realistic behaviour. In the present work a model is proposed considering aspects of Thornton and Ning’s [18], Tomas’s [19], and Walton and Johnson’s [21] models. Sensitivity analyses of the proposed model parameters on bulk compaction behaviour are also reported.

In the following an overview of the available linear models reported in the literature for elasto-plastic and adhesive contact deformation is presented.

*Luding’s model* [20]:

Luding’s model contains a shortcoming by which the behaviour of elasto-plastic and adhesive contacts is not realistically simulated. Contacts “break” at zero overlap (\(\alpha = 0\)), regardless of loading or unloading history. This implies that plastic deformation has been ignored, which is unrealistic since plastic deformation is permanent and hence detachment must take place at \(\alpha > 0\).

*Walton and Johnson’s model* [21]:

## 2 The proposed contact model

*BC*, for which the contact force is given by Eq. (5),

*BC*line. Unloading beyond point \(B\) is governed by a “stiffness” \(-k_{e}\). The contact breaks at a negative overlap, \(\alpha _{fe}\), at point \(A\) with the contact force being \(5f_{ce}/9\).

*CD*.

*EF*), a negative elastic stiffness, \(-k_{e}\), is considered. The governing equation for this part of contact force can be evaluated by Eq. (8),

### 2.1 The load-dependent pull-off force

The magnitude of pull-off force increases with plastic deformation initially very rapidly and then more linearly, an observation which is also present in the elasto-plastic and adhesive models of Thornton and Ning [18], Tomas [19], Luding [20], and Walton and Johnson [21]. \(\alpha _{cp}\) is directly related to \(\alpha _{p}\), for which a similar functional relationship prevails.

### 2.2 Impact, rebound and critical sticking velocities

The coefficient of restitution reaches an asymptote at very high impact velocities. The asymptotic value is a function of contact properties. Increasing the interface energy results in a reduction of the coefficient of restitution since the cohesion is increased. Increasing the elastic stiffness also reduces the coefficient of restitution, since the plastic work is increased. Increasing the plastic stiffness increases the coefficient of restitution, since the contact becomes stiffer and the extent of plastic deformation is reduced.

### 2.3 Linearization of the locus of pull-off force

*BC*), so that

*CD*), a negative elastic stiffness, \(-k_{e}\), is considered, leading to the contact force

## 3 Comparison of the proposed model with that of Thornton and Ning [18]

Properties of ammonium fluorescein particle and silicon wall used in Ning’s [3] simulations

Property | Particle | Wall |
---|---|---|

Radius (\(\upmu \hbox {m}\)) | 2.45 | – |

Density \((\hbox {kg/m}^{3})\) | 1,350 | 1,350 |

Elastic modulus (GPa) | 1.2 | 182 |

Poisson’s ratio (\(-\)) | 0.3 | 0.3 |

Interface energy \((\hbox {J/m}^{2})\) | 0.2 | |

Contact yield pressure (MPa) | 35.3 |

Model parameters obtained by determining the slopes of the responses in Fig. 11

\(k_{e}\) (N/m) | 1,500 |

\(k_{p}\) (N/m) | 210 |

\(k_{cp}\) (N/m) | \(-\)20 |

\(f_{0}\,(\upmu \hbox {N})\) | \(-\)2.1 |

\(f_{0p}\,(\upmu \hbox {N})\) | \(-\)4.0 |

Comparing Figs. 11 and 12 a good agreement is observed between the simplified proposed model and that of Thornton and Ning in terms of the maximum overlap and permanent plastic deformation at different impact velocities.

Figure 13 shows a reasonable agreement between the proposed model and that of Thornton and Ning given the simplification. The dotted lines are obtained analytically by applying an energy balance as described in Sect. 2.2. Since the simulation results of the proposed model lie on the analytical curves, it can be concluded that the energy balance considerations in Sect. 2.2 are correct. The critical sticking velocity in the proposed model is very similar to that of Thornton and Ning (\(\sim \)1.6 m/s). With the proposed model, the coefficient of restitution increases initially, however it reaches an asymptotic value of \(\sim \)0.38 for impact velocities larger than about 10 m/s. The reason for this is attributed to a constant elastic stiffness, for which the ratio of elastic work \((W_{e})\) to plastic work \((W_{p})\) is always constant. At very high impact velocities, the elastic strain energy becomes very large relative to the work of adhesion \((W_{ad})\), so the elasto-plastic process dominates since the plastic-cohesive stiffness, \(k_{cp}\), is normally smaller than the plastic stiffness, \(k_{p}\). Hence the asymptote has a value of (\(k_{p}/k_{e})^{1/2}\) [14]. This asymptotic behaviour is not in line with the experimental evidence [27, 28], where it is shown that at sufficiently high velocity the impact energy far exceeds the adhesion energy and the coefficient of restitution is primarily a function of the energy loss due to plastic deformation.

As expected a decreasing coefficient of restitution is observed at high impact velocities, although showing a deviation form Thornton and Ning’s model.

New experimental data are needed to check the validity of the model’s predictions, as a larger data set than that available for ammonium fluorescein particles would provide more certainty.

There are clear differences between this model and those of Walton and Johnson [21] and Luding [20] in (1) the point of contact detachment, (2) reloading on the adhesive branch and (3) most importantly in the work of adhesion. The latter strongly influences the sticking velocity, as it depends on the balance between elastic strain energy and work of adhesion. Hence a rigorous test of these models is to experimentally verify the sticking velocity of materials, whose model parameters have been independently characterised.

## 4 Sensitivity analysis of the proposed model parameters

Model parameter values used in the simulations

Parameter varied | \(k_{p}\) (kN/m) | \(k_{e}\) (kN/m) | \(k_{cp}\) (kN/m) | \(\varGamma \,(\hbox {J/m}^{2})\) |
---|---|---|---|---|

\(k_{e}\) | 10 | 50, 100, 500, 1,000 | 0 | 0 |

\(k_{p}\) | 10, 50, 100, 250, 500 | 1,000 | 0 | 0 |

\(k_{p}\) | 50, 100, 500, 1,000, 2,500 | 5,000 | 0 | 0 |

\(k_{p} = k_{e}\) | 100 | 100 | 0 | 0 |

\(\varGamma \) | 100 | 1,000 | 5 | 0.05, 0.1, 1, 2, 5 |

Size distribution of the generated particles

Particle diameter (mm) | 0.8 | 0.9 | 1 | 1.1 | 1.2 |

Number frequency (%) | 5 | 25 | 40 | 25 | 5 |

Large stiffness ratio values imply particles deforming extensively plastically, whereas a stiffness ratio of one implies a purely elastic deformation. For the stiffness ratio of one, the plastic component of the work is still larger than the elastic one. The plastic work in this case is due to particle rearrangements and frictional dissipation between the particles themselves and with the walls, since the normal contacts deform elastically. The graph shows that as the stiffness ratio increases, the fraction of plastic work increases, while that of elastic work decreases. The increase in the ratio means either the plastic stiffness is decreased or the elastic stiffness is increased. If the plastic stiffness is decreased while the elastic stiffness is kept constant (softer particles), more work is expended in deforming contacts to reach the same force. This leads to an increase of the total work, while the elastic work remains the same. Therefore normalized elastic work decreases and normalized plastic work increases accordingly. In the case where the elastic stiffness is increased while plastic work is kept constant, the total input work does not change, but the fraction of elastic work decreases. This leads to a decrease in the normalized elastic work and consequently normalized plastic work increases. It can also be seen from Fig. 16 that there exists a limit for the stiffness ratio (\(k_{e}/k_{p}\approx 20\)) beyond which almost all of the work input into the system is used in plastic deformation.

## 5 Conclusions

A new linear elasto-plastic and adhesive contact model for spherical particles has been proposed based on the models of Luding [20] and Walton and Johnson [21] and considering aspects of Thornton and Ning’s [18] and Tomas’s [19] contact models. Plastic deformation of contacts during loading and pure elastic unloading, accompanied by adhesion are considered, for which the pull-off force increases with plastic deformation and for which the detachment is governed by the work of adhesion. Considering the collision of an adhesive spherical body with a rigid flat target, the critical impact velocity above which rebound occurs, as predicted by the proposed model, is found to be very similar to that of Thornton and Ning’s model. This agreement is improved by considering a load-dependent unloading stiffness.

Sensitivity analyses of the model parameters on work of compaction reveal that by increasing the stiffness ratio (\(k_{e}/k_{p}\)) the normalized plastic work increases and the normalized elastic work decreases. By increasing the interface energy, the plastic work increases, however the elastic work does not change notably. This highlights the flexibility of the model in representing the mechanical behaviour of a wide range of particulate materials. The linear nature of the model leads to time efficient simulations, whilst still capturing the complex material behaviour.

## Notes

### Acknowledgments

The financial support of the Engineering and Physical Sciences Research Council, UK, through the Grant EP/G013047 is gratefully acknowledged. The authors are also grateful to the constructive critical comments of Professor Stefan Luding and the reviewers.

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