# Numerical and experimental investigation of an elastoplastic contact model for spherical discrete elements

## Abstract

A contact model for the normal interaction between elastoplastic spherical discrete elements has been investigated in the present paper. The Walton–Braun model with linear loading and unloading has been revisited. The main objectives of the research have been to validate the applicability of the linear loading and unloading models and estimate the loading and unloading stiffness parameters. The investigation has combined experimental tests and finite element simulations. Both experimental and numerical results have proved that the interaction between the spheres subjected to a contact pressure inducing a plastic deformation can be approximated by a linear relationship in quite a large range of elastoplastic deformation. Similarly, the linear model has been shown to be suitable for the unloading. It has been demonstrated that the Storåkers model provides a good evaluation of the loading stiffness for the elastoplastic contact and the unloading stiffness can be assumed as varying linearly with the deformation of the contacting spheres. The unloading stiffness can be expressed in a convenient way as a function of the Young’s modulus and certain scaling factor dependent on the dimensionless parameter defining the level of the sphere deformation.

## Keywords

Contact Elastoplastic Spheres Discrete element method Unloading## 1 Introduction

Nowadays, the discrete element method (DEM) is used for modelling various particulate and nonparticulate materials such as soils, rocks, ceramics. The material in the DEM is represented by a large assembly of particles interacting among one another with contact forces. The particles can be of arbitrary shape, but spherical particles are often chosen because of the simplicity and computational efficiency of the numerical algorithm. The spherical particles will be considered in the present study.

The normal contact between particles in the DEM is very often modelled assuming an elastic-type force–displacement relationship. A simple contact model consisting of a linear spring in parallel with a viscous damping element proposed in the pioneering work by Cundall and Strack [4] is still very popular in DEM simulations [3, 7, 16]. A nonlinear elastic interaction is included in the contact models based on the Hertz theory [5, 11, 25].

In many applications, however, particle deformation due to contact cannot be treated as purely elastic. Because of the contact force concentration, yielding at the contact zone between two spheres made from ductile materials, e.g. metals, may occur at a relatively low loading [17]. In such cases, a partial irreversibility of interparticle penetration should be included in the contact model in the DEM. Various elastoplastic models have been proposed for the DEM, e.g. [10, 15, 20, 21, 22, 23, 24].

The model proposed by Thornton [22] considers both the elastic and plastic ranges of the contact interactions. Initially, the contact is considered assuming the Hertzian elastic model. The loading is switched to the plastic regime when the yield criterion is satisfied. The yield criterion used by Thornton [22] is based on the assumption that the contact stress distribution after yielding can be obtained by ”cutting off” the Hertzian stress distribution. The Thornton’s model is used in different applications which require consideration of plastic effects in the interparticle interaction, e.g. [2]. It has been, however, found out that the Thornton’s model underestimates significantly the contact force in the range of plastic loading, cf. [15, 23]. Improved elastoplastic contact models analogous to that of Thornton have been proposed by Rathbone et al. [15] and by Vu-Quoc et al. [23]. In both cases, the improvements were based on the results of the finite element analyses of the contacting spheres.

The range of elastic loading is often very small in comparison with subsequent plastic loading, cf. [17]. Then, it can be neglected and the contact deformation can be treated as plastic from the contact initiation. This is consistent with the rigid-plastic or rigid-viscoplastic material model. A model assuming rigid-plastic behaviour according to the Hollomon stress–strain-hardening curve has been developed by Storåkers et al. [20, 21]. The Storåkers contact model has been successfully used for discrete element modelling of powder compaction [12].

An issue of great importance in the elastoplastic contact models is the choice of a suitable unloading model. The Hertzian elastic unloading with a modified contact curvature proposed by Thornton [22] is usually assumed in the elastoplastic contact models, e.g. [15, 23].

As it has been shown by Pasha et al. [14], a rigorous treatment of the elastoplastic contact deformation is often not necessary to represent behaviour of the granular material, therefore simple linear elastoplastic models such as that proposed by Walton and Braun [24] are still important for many practical applications. The elastoplastic loading and elastic unloading in the Walton–Braun model are governed by linear force-displacement relationships with different slopes, which ensures a residual overlap of the particles when the contact force drops to zero. No tensile forces are allowed in the Walton–Braun model. The elastoplastic model developed by Luding [10] can be considered as an extension of the Walton–Braun model to the adhesive contact.

Despite extensive research and apparent simplicity of the problem, the question of applicability ranges and accuracy of linear elastoplastic contact models in the DEM is not fully answered. The present work is aimed at numerical and experimental investigation of validity of the linear Walton–Braun-type elastoplastic model and suitability of analytical formulae for evaluation of model parameters defining the loading and unloading stiffness.

Discrete element modelling of powder metallurgy processes, including compaction, sintering and cooling is a practical motivation of the present work, therefore the formula proposed by Storåkers for powder compaction [8, 13] will be verified for evaluation of the loading stiffness. A special interest will be paid to the unloading behaviour. The linear approximation of the unloading relationship will be checked against numerical and experimental data. The linear contact model in the DEM is consistent with linear macroscopic relationships, e.g. [9]. A linear unloading model is searched as a suitable microscopic counterpart of the linear thermoelastic macroscopic description of the unloading of the sintered material during cooling.

## 2 Formulation of the elastoplastic contact model

### 2.1 Problem formulation

*i*and

*j*with radii \(R_i\) and \(R_j\) (Fig. 1) is considered. The particles’ positions are defined with the position vectors of their centroids, \(\mathbf{x}_i\) and \(\mathbf{x}_j\). The contact force exerted on the

*i*-th particle by the

*j*-th is denoted by \(\mathbf{F}_{ij}\), and by the Newton’s third law the contact force exerted on the

*j*-th particle \(\mathbf{F}_{ji}\) satisfies the relation:

*i*:

*h*(Fig. 1):

*h*is used for the determination of the contact force. Please note that the convention adopted for the vector

*n*and the overlap

*h*is such that the contact compressive force

*F*is positive and so is the overlap

*h*when the particles are in contact.

### 2.2 Walton–Braun model

*S*and

*B*are certain constants.

### 2.3 Elastoplastic loading model

*i*-th and

*j*-th particle materials which are defined by the Hollomon stress–strain relationships

*m*is the hardening exponent. The normal interaction force

*F*is given by the following equation:

*h*is the particle overlap given by Eq. (4), the parameter \(c^{2}\) is given by [6]:

### 2.4 Elastic unloading model

*i*and

*j*.

## 3 Experimental results

### 3.1 Compression of steel balls

*h*/ 2

*R*, respectively. The force–displacement curves scaled in this way are plotted in Fig. 9. It can be observed that the scaled curves for the balls of different sizes merge pretty well. This shows that the scaled relationships can be used for different particle sizes provided that the material properties are the same.

### 3.2 Determination of stress–strain curves

## 4 Finite element method simulation

The contact of an elastoplastic sphere against a rigid plane surface has been simulated using the finite element method. The analysed problem can be considered as quasistatic and it can be solved using either static or dynamic formulation. The explicit dynamic FE framework which is commonly used for metal forming processes [18, 19] has been chosen here. The isotropy of material properties has been assumed. The geometry, loading and boundary conditions are symmetric about an axis of rotation; therefore, the problem can be solved using two-dimensional finite elements keeping the features of the three-dimensional description.

A 2D axisymmetric model has been prepared for the ABAQUS/Explicit FE program [1]. A hemisphere with a radius of 15 mm has been discretized nonuniformly with 3175 four-node bilinear axisymmetric quadrilateral CAX4R finite elements with the reduced integration and hourglass control. The finite element model is shown in Fig. 13. A magnified detail in Fig. 13b illustrates a mesh refinement in the contact area where high stress and strain gradients are expected. Frictionless contact conditions have been assumed between the hemisphere and the plane. Compressive loading has been introduced by applying the vertical displacement to the top plane surface of the hemisphere without restraining the horizontal motion. Similarly, the unloading has been controlled kinematically. In order to ensure quasistatic conditions, the loading and unloading have been applied using sinusoidal velocity profile with a sufficiently low-velocity amplitude. Maximum velocity 0.1 m/s has been taken for the loading and 0.02 m/s for the unloading.

*C*, \(\varepsilon _0\) and

*n*are material parameters. The Swift curve parameters have been obtained by fitting the numerical stress–strain curve to the experimental one given in Fig. 12. Comparison of the analytical curve defined by the Swift parameters: \(C=809\hbox { MPa}\), \(\varepsilon _0=0.0013\) and \(n=0.232\) with the experimental stress–strain curve is shown in Fig. 14.

The numerical and experimental scaled contact force–displacement curves are plotted in Fig. 15. It should be noted that the numerical curve in Fig. 15 has been constructed taking *h* equal twice the applied displacement to the plane of the hemisphere in order to use consistent parameters with those defined in Fig. 6b. Both curves plotted in Fig. 15 are very close to each other which confirms correctness of the numerical model. It can be noted, however, that while the slopes of the experimental and numerical loading curves are nearly the same, the difference in the slopes of the unloading curves is significant. This will be discussed in more detail further on when parameters of the unloading model will be evaluated.

## 5 Validation of the analytical contact model for loading

The analytical Storåkers contact model presented in Sec. 2 will be validated against the experimental and simulation data. Figure 18 shows the scaled force–displacement relationship according to the linear Storåkers contact model compared to the experimental and simulation curves.

The decrease in stiffness manifesting itself by the deviation of the experimental and numerical curves from the Storåkers model at higher values of *h* / *R* is due to large deformations of the compressed balls.

## 6 Analysis of the unloading model

*h*/ 2

*R*, respectively, during unloading are related linearly:

The plots in Figs. 20 and 21 reflect the difference between the unloading slopes in the experimental studies and numerical simulations which can be observed in Fig. 15. Experimental studies have been performed taking care to compensate properly the machine stiffness in order to eliminate its influence on the results. It was impossible, however, to avoid an error due to an indentation of the compressed balls into the plates. These problems do not occur in the FEM simulations; therefore, as far as the unloading stiffness is concerned, the numerical results are more credible than the experimental ones obtained in the tests performed in this work.

*B*,

*b*and \(\beta \) define the variation of the unloading modulus with an increasing deformation, and \(k_L\), \(\kappa _L\) and \(\alpha _L\) define the loading stiffness. These parameters determined from experimental and numerical data are given in Table 1.

Parameters of the unloading models approximating numerical and experimental data

Parameter | FEM | Experimental |
---|---|---|

\(k_L\) (MN/m) | 43.263 | 43.263 |

\(k_U^{0}\) (MN/m) | 266.72 | 364.06 |

| 439745.6 | 121328.2 |

\(\kappa _L\) (MN/m) | 1442.1 | 1442.1 |

\(\kappa ^{0}_U\) (MPa) | 8890.7 | 12135.4 |

| 439752.0 | 120692.2 |

\(\alpha _L\) | 8.879E-03 | 8.879E-03 |

\(\alpha ^{0}\) | 0.0565 | 0.0729 |

\(\beta \) | 2.693 | 0.7501 |

## 7 Conclusions

The analysis of the experimental and numerical results has confirmed some known observations as well as it has given a new insight into the inelastic contact between spheres which can be useful to model the interparticle contact in the discrete element method. Both experiments and numerical results have shown that the elastic model has a very limited validity if plastic effects should be taken into account. The plastic deformation is initiated at a low load and the contact force cannot be predicted using the elastic contact models, for instance the Hertz one. Both numerical and experimental results have shown that the force–displacement curve can be approximated by a linear relationship in quite a large range of elastoplastic deformation of the contacting spheres. It has been shown that the linear Storåkers model predicts the force–displacement relationship which agrees very well with the experimental and numerical results.

Much of the attention in the present work has been paid to the modelling of the unloading in the contact between inelastic spheres. It has been demonstrated that the unloading can be approximated accurately by the linear model with the modulus varying linearly with the maximum deformation achieved in the contact. Based on the analysis of the experimental and numerical data, it has been proposed that the initial unloading stiffness can be larger than the loading stiffness. The parameters for the unloading model has been evaluated. Use of the size-independent parameters to describe the model allows us to assume that the results can also be used for other materials and particle size in the discrete element method.

The presented results and their analysis have shown that the Walton–Braun-type model with the loading stiffness evaluated according to the linear Storåkers model and the linear elastic unloading with linearly varying unloading stiffness should be an efficient and accurate model for the inelastic contact in the discrete element method using spherical particles. Obviously, one should be aware of the limitations of this model.

## Notes

### Compliance with ethical standards

### Conflicts of interest

The authors have declared that no conflict of interest exists.

### Human and animal participants

This article does not contain any studies with human participants or animals performed by any of the authors. Informed consent was obtained from all individual participants included in the study.

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