Application of the discrete element method to model cohesive materials

Timo Gaugele1 ∗, Michael Storchak2, and Peter Eberhard1

1 Institute of Engineering and Computational Mechanics, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart,Germany.

2 Institute for Machine Tools, University of Stuttgart, Holzgartenstrasse 17, 70174 Stuttgart, Germany.

In this contribution we show how the discrete element method (DEM) can be applied to model cohesive materials whichexhibit ductile behaviour by introducing connective elements that can bear a certain load. The modelled material is verifiedby simulating a orthogonal cutting process which is compared to experimental results.

1 Introduction

One way to include fracture and fragmentation when analyzing loaded mechanical systems is based on the DEM, [2], [3]. Inrecent years this approach was adopted in geomechanics and civil engineering to model fracture in elastic-brittle materials,e.g. [1] and [5]. Using this approach, the material is considered as being fully discontinuous and made up by assembling andbonding adjacent discrete elements. The particle bonds are represented by force laws and can sustain only a specified stressuntil failure.

2 Granular solid

Different types of force laws can be used depending on the considered problem to be modelled as well as the type of particlesused [1]. In this paper, we focus on spherical particles bonded by rods. As the used rods can only bear tensile loads and therods are fixed in the center of mass of each particle there is no need to introduce rotational degrees of freedom for the spheres.The force laws representing the rods contain a viscous part as well as an elastic-plastic part as ductile materials are to bemodelled, too. The elastic-plastic part of the force law used in our model is based on a piecewise linear hardening model, [4],and is depicted in Fig. 1. It is explained in detail in [3]. Characteristic parameters of this model are the initial yield limit ε0.2,Young’s modulus E and the slope 0 < k < E of the yield function beyond the yield point σ0.2.

For a material without previous plastic deformation it is assumed that the stress-strain relationship is governed by a piece-wise linear function σ = F (ε) which is symmetrical about the origin, i.e.

σ(ε) = −σ(−ε). (1)

If the considered specimen is loaded to σy beyond the elastic limit σ0.2 the material flows plastically. To include theBauschinger effect it is understood that the plastic deformation shifts the yield function in the σ−ε-plane. The shift parameters

σ0 and ε0 can be found by setting g(ε0)!= h(ε0), see [3]. As a result the yield function (1) is transformed to

F̂ = σ(ε) = σ0 + F (ε − ε0). (2)

The stress then can be readily calculated as

σ = σ0 + E(ε − ε0). (3)

As known from experience a loaded specimen does not flow infinitely but fails once stress reaches tensile strength. Inorder to reproduce realistic behavior, the tensile strength σmax is introduced in the model. Consequently, if σ > σmax the rodconnecting two considered particles is removed. In case of contact of particles which are not connected via bonds the contactforces are governed as described in [3].

In order to apply the method to real-life problems parameters have to be adequately determined. Here it is chosen tocompare the results of a DEM-model and a FEM-model for a given scenario where the loading of a structure in a quasi-static manner is considered. As a result the stress-strain relationships of both models can be compared and if necessary theparameters of the DEM model can be adjusted.

To show the applicability of the DEM to model cohesive materials the case of orthogonal cutting is considered with specialemphasis on cutting forces. As a pre-processing step, the solid representing the workpiece is generated as a bulk of identical

∗ Corresponding author E-mail: gaugele@itm.uni-stuttgart.de, Phone: +49 711 685 66388, Fax: +49 711 685 66400

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

PAMM · Proc. Appl. Math. Mech. 7, 4010013–4010014 (2007) / DOI 10.1002/pamm.200700611

σy

σ0.2

σ0

εεyε0

σ

h(ε)

g(ε)

Fig. 1 Stress-strain curve for plasticity law.

0 0.01 0.02 0.03−200

0

200

400

600

DEM

planingexperimental

cutti

ng f

orce

[N

]

displacement [m]

Fig. 2 Experimental and simulated cutting force.

Fig. 3 Snapshots of a simulation of orthogonal cutting of elastic-plastic material. The fraction of broken rods is color coded.

spheres arranged in a regular face-centered lattice. All adjacent spheres are bonded using the force laws described above. Theworkpiece is machined using a tool represented by triangles and moved according to a function of time. If the tensile strengthof the material is reached the rods representing the cohesive forces of the material are removed and cannot be restored. Somesnapshots of intermediate states are shown in Fig. 3. No friction is considered in this scenario.

To test the validity of the model we compare the results of the cutting force acting on the tool with results made in planingexperiments with aluminum shown in Fig. 2. The model parameters are chosen as E = 7e10N/m2, k = 1.4e9N/m2,ε0.2 = 0.0045 and σmax = 6.5e8N/m2. Further parameters are the uncut chip thickness given as 0.9mm, workpiecedimensions 15mm x 7mm x 1.3mm, a feed rate of 1.6m/s, and a particle radius of 0.2mm which results in 1909 particles.The forces shown in Fig. 2 refer to a width of cut of 1mm. The cutting force calculated with the DEM rises sharply to aplateau when the tool hits the workpiece. The experimental result differs from that in the beginning because of slackness ofbearing but reaches a similar value in the end where both forces show good agreement.

3 Conclusion

The DEM can be used to model fracture in ductile materials in cutting processes which is shown by good agreement withexperimental results.

Acknowledgements The work on cutting processes is funded by the DFG in the framework of the SPP1180. All calculations have beenmade using PASIMODO by F. Fleissner, ITM, University of Stuttgart. All support is highly appreciated.

References

[1] G.A. D’Addetta and E. Ramm. A microstructure-based simulation environment on the basis of an interface enhanced particle model.Granular Matter, 8, 159–174 (2006).

[2] M.P. Allen and D.J. Tildesley. Computer Simulations of Liquids. Clarendon Press, Oxford, 1989.[3] F. Fleissner, T. Gaugele, and P. Eberhard. Applications of the Discrete Element Method in Mechanical Engineering. Multibody System

Dynamics, 18, 81–94 (2007).[4] K. Liu, L. Gao, and S. Tanimura. Application of discrete element method in impact problems. JSME International Journal, Series A,

47(2), 138–145 (2004).[5] A.V. Potapov, M.A. Hopkins, and C.S. Campbell. A two-dimensional dynamic simulation of solid fracture. International Journal of

Modern Physics C, 6(3), 371–398 (1995).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

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